I have a question on quasicategories which seems to be unavoidably cubical, and which I haven't been able to locate any information about. Suppose I have a cube $U:\mathbf{2}^3:\to C$ in a quasicategory $C$, with, say, its front and back faces $Y,Y':\square\to C$, its left and right faces $X,X'$, and its top and bottom faces $Z,Z'$. Let $W$ be another square which I want to replace $Y$ with. Given that $W$ and $Y$ induce the same morphism in $Ho(C^\mathbf{2})$, I can do this replacement without changing $Y',Z,Z'$ by working in $C^\mathbf{2}$, since I have a homotopy commutative square $ZY'=Z'W$ which I can always rectify to an actual square.
I think I can also replace $Y$ with $W$ while saving $X$, just by filling the cubical "horn" $X,W,Y',Z,Z'$ 3-simplex by 3-simplex. But this will still generally change $X'$.
Question: Is there any condition I can put on the two squares $Y,W$ in my quasicategory $C$ that ensures I can exchange $W$ for $Y$ in a cube without changing any other faces?