"Bosis" on Shabbos

Thank you for your question.

It does not become Bosis, since it is written in the Shulchan Aruch, that one cannot make something Bosis if it is not his without the consent of the ba'al habayit. This is the quote of the Rema;

שולחן ערוך אורח חיים הלכות שבת סימן שט סעיף ד
אם אדם הניח דבר מוקצה על של חבירו, לא אמרינן דנעשה בסיס לדבר האסור, דאין אדם אוסר של חבירו שלא מדעתו (א"ז). 

Shulchan Aruch, Orach Chaim – Laws of Shabbos, Siman 309, Se’if 4

If a person placed a muktzeh item on top of his friend’s object, we do not say that the object becomes a basis for a forbidden item (basis le’davar ha’assur), because a person cannot render his friend’s property forbidden without his knowledge. (Ohr Zarua)

Wishing you much success.