Should one wear tefillin for tashlumin for Shacharis?

Question

Let's say Aharon missed Shacharis. He needs to pray Shemoneh Esrei twice by mincha time to make it up, which is called "tashlumin" (תשלומין). (OC 108:1)

Now, Aharon can put on tefilin at any point in the day (though he won't do so at night).

Is there any reason to put them on specifically when he davens (prays) his tashlumin (the second prayer)? After all, that prayer is a stand-in for Shacharis.

I'm looking for an answer that says "this sefer says" or the equivalent; I'm also capable of saying "hey, can't hurt, right?"

Answer 1

Rabbi Rachamum Mazuz quoting Yalkut Yosef 1:78 in a case where one missed Shacharus and Musaf on Rosh Chodesh says that first he should put on Tefilin and remove them, then Daven Mincha, then Daven Musaf, and then do Tashlumim for Shacharis.

קודם כל יניח תפלין בברכה ויוריד אותו, ואח"כ יתפלל מנחה, ואחריה תפלת מוסף, ואח"כ תשלומין של שחרית

The way I understand this ruling: If there was a reason to specifically put them on for the Shacharis Tashlumin, why not answer first Daven Mincha, then Daven Musaf, then put on Tefilin and Daven the Shacharis Tashlumin? However he does not say so, leading me to believe it is not necessary nor is there any reason to do so.

However Birur Halacha 108 quoting Rabbi Moshe Nechemia Kahanow Z"L says that one should put on Tefilin when he is Davening Mincha, and keep it on for the Tashlumim if he has not put it on yet that day.