I have a continuous function $f : X \rightarrow X$, where $X$ is an $8$-dimensional simplicial complex. I want to homotope $f|_{\text{skel}^2(X)} : \text{skel}^2(X) \rightarrow X$ to a function $g : \text{skel}^2(X) \rightarrow X$ so that $\text{im}(g) \subseteq \text{skel}^2(X)$. Is this possible? Would it still be possible if I also required the homotopy to fix points that are already in the $2$-skeleton?
If this is possible, maybe it could be accomplished by somehow "pushing" points lying in higher dimensional faces to the boundary?