Your question is supposing that without door, The immobile component is not Chayav Mezuza. See here the components of a door. Strike Jamb and Hinge Jamb are called mezuzoth. The door itself is a priori not concerned.
Rambam and Raavad (In Mishne Torah Sefer Ahava Hilchoth Tefilin Umezuza, Chapter 6 Halacha 5) are controversy about the meaning of a Gemara
This results in that the former requires the presence of a door, the second requires simply that there are hinges.
You are asking following the first (but The Shulchan Aruch seems to prefer the other opinion)
The Kessef Mishne cit it a response from the Rambam sages of Lunel,
The important thing is that there opportunity to close the door (by the way I think the chute system has a hinge of status after the Halacha).
According to the Rambam
The sliding type's door, with the door is Chayav Mezuza ,
and according to the Raavad,
it is Chayav also with the chute only, in the absense of door.