In some conditions he can leave, in some else he becomes Shomer Aveda. Let's look Chazal and poskim.
1.Mitsvat Hashavat Aveda remains for ever.
The Mishna BM 28b stated that a part of lost objects which produces and consumes, will continue to produce and consume. Something which doesn't will be sold. Gemara asked, according to Rashi understanding, "needs he struggle for ever to watch the lost object?" Basically, the bottom line is that after a given period, you can sell it and keep the money for ever, according to Rabbi Tarfon he can even borrow this money. We learn this from a verse from which Gemara learned "consider how to return it unto him".
But to leave at all is contrary to the mitsvat hashava, which is for ever i a certain form.
Thus, a lost object has a Mitsvat hashava for ever. But this is still different from the OP object in several points: the owner of the aveda has a good excuse for his delay, he probably didn't find where is the object. But he perhaps wived. The last point (called "yeush") does not lead to a kula because of the mitsva to return which already began before yeush. This person needs help and the Tora give it.
2.Custody becomes sometimes Hashavat Aveda.
For a custody, the delay may be expression of a give it to the custodian, or can be a sign that the owner has an "force majeure" and became a kind of baal aveda. In summary in shomer some cases may be interpreted as voluntary abandon by the owner, which makes the shomer patur, but not all cases.
Mishna in third chapter of BM can help a lot.
If a man deposits produce with his neighbour, even if it is suffering loss, he must not touch it. r' simeon b. gamaliel said: he must sell it by order of the court, because it is like returning lost property to its
We can assume this the custodian is not in position to alert the owner. It is a bit similar to the OP case, the owner should have to come back and look for his fruits. To leave the fruits rotting is a lesser degree than stopping custody, but it is stiff prohibited. Here appears the "hashavat aveda issue". We ruled halacha as Raban Shim'on ben Gamliel.
Pesachim 13a:
For Rabin son of R' Adda related: It once happened that a certain man deposited a saddle-bag full of leaven with Johanan of Hukok, and mice made holes in it, and the leaven was bursting out.
He then went before Rabbi.
The first hour he said to him, 'Wait'; the second, he said to him, 'Wait'; the third he said to him, 'Wait'; the fourth he said to him, 'Wait'; at the fifth he sa to him, 'Go out and sell it in the market'.
This case is almost the case of the OP, why the owner of the Chamets didn't come back? Gemara compared this case to the above for the sentence of Rashbag.
3.The custodian is not the slave of the bailor.
But there are cases of leniency in custody if we know that the owner is located in the same city and doesn't care about his object. For instance, for some objects one must air them regularly, as books or clothes. If the owner is no far away from the place, the custodian is exempt from airing the object. See Rambam Sheela 7, 4
Finally perhaps in 7, 12, the Rambam addresses the OP :
A question arose when a person entrusts an article to a colleague and then journeys overseas, and afterwards, the watchman also desires to travel overseas or depart in a caravan. There is an authority who ruled that if the watchman brings the entrusted article to the Jewish court, he is absolved of his responsibility. These are well-reasoned words. For we do not imprison the watchman in this city because of the object entrusted to him by the person who departed overseas . The watchman cannot take the entrusted article with him, lest it be destroyed by factors beyond his control. The court should then entrust the article to a faithful person. This is like returning a lost object to its owner.
This case is very linked which our case, but if the owner is located in the same city and the watchman desire to travel, obviously he can say it "I leave". If there is no way to find the owner, the case of hashavat aveda apply. But the duty of Hashavat Aveda is not only for the custodian, I assume that for hashavat aveda too, a similary din exists.
The source is in BK 118a:
MISHNAH:
If a man has robbed another, or borrowed money from him, or received a deposit from him in an inhabited place, he may not restore it to him in the wilderness; [but if the transaction was originally made] upon the stipulation that he was going into the wilderness, he may make restoration even while in the wilderness.
GEMARA:
A contradiction could be raised [from the following:] 'A loan can be paid in all places, whereas a lost article [which was found], or a deposit cannot be restored save in a place suitable for this'? - Said Abaye: What is meant is this: 'A loan can be demanded in any place, whereas a lost article [which was found] or a deposit cannot be demanded save in the proper place.
Concerning a time limited watching, I found something in Sefer Machane Efrayim Hilchot Shomerim, siman 18.
The ME said that following Tos. in Meruba and Rashba in Kiddushin 13, Ran in Hashoel, stated than Shomer Sachar and Shomer Chinam can waive their function even before the end of the shmira, and according to Ritba who is stringent, at least at the end of a shmira limited in time, he can say that he stop the Shmira. {He can make a kind of hachraza to make sure that the owner knows that he will end the shmira}.
But the question is if there is nobody to say this to him.
I believe that the absence of the owner must be interpreted as voluntary if he remained here. But as an Ones if he is far away , the Shomer becomes Shomer aveda.
Halacha:
SA CM 293, 3:
המפקיד אצל חבירו והלך בעל הפקדון למדינת הים והנפקד רוצה לפרש בים או לצאת בשיירא אם בא והביא הפקדון לב"ד נפטר מאחריות שמירתו שאין אוסרים זה במדינה זו מפני פקדון של זה שהלך [ואי אפשר] (וא"א) לו להוליכה עמו שמא יארע לו אונס ויהיה חייב באחריותו ב"ד מפקידים ביד אדם נאמן אצלם מפני השבת אבידה לבעלים:
הגה: ואם לא היה יכול להניחו בישוב והוצרך להוליכו עמו למדבר מכח סכנה או מכח שומר שמסר לשומר ונאנס מידו י"א דפטור הנפקד (מרדכי פ' הגוזל):
קצות החשן ס"ק ב': ... כל אחריות עליו... עד הזמן שנתחייב בדין שומרים...
He is not required to continue to watch, and can travel.
He must seek assistance from Bet Din for a problem of hashavat aveda.
If the watching is time limited the duty to watch ends with this time (Qtsot, but maybe that everybody agrees), but the Sheela of Hashavat aveda is not eliminated.
