I was not sufficiently clear on my last attempt at asking a similar (but not identical) question. Tom Goodwillie mentioned (in the accepted answer) that the question can be reduced to this one and mentioned a fact of which I was unaware, but after trying to prove this for the past day, I have returned to ask for a sketch of the proof:

Let $X$ be a simplicial set, and let $p:X\to \Delta^n$ be a right fibration (has the right lifting property with respect to all right horn inclusions (equivalently, it has the right lifting property with respect to the maps $$\Delta^1\times A\coprod_{\{1\}\times A}\{1\}\times B\to \Delta^1\times B$$ for any inclusion $A\hookrightarrow B$). (These generate the same weakly saturated class of maps)).

Let $i_0:\{0\}\hookrightarrow \Delta^n$ be the inclusion of the 0th vertex of $\Delta^n$. We'd like to show that the pullback of this map by $p$ (i.e. the induced map $f:X\times_{\Delta^n} \{0\}\hookrightarrow X$) is a deformation retract.

According to Tom, we can use the second characterization of right fibrations to obtain some kind of lifting of the homotopy and retraction for $i_0$ (which is a deformation retract).

Would somebody mind sketching the proof?

Edit: If this question is answered by 2AM EST (roughly three hours and forty-five minutes from the time of this current edit), I will award the answerer with a 450 point bounty as soon as it becomes possible (one must wait 48 hours from when the question was asked).