Let \begin{array}{ccc} X & \xrightarrow{} & Y \\ \downarrow & & \downarrow\\ Z & \xrightarrow{} & W\\ \end{array} be a homotopy commutative diagram in a proper model catgeory $\mathcal{C}$ with $P$ be the homotopy pushout of the diagram $Z \leftarrow X \to Y.$
Consider the following (homotopy)commutative diagram
with $Y' \to Y, Z'\to Z, W'\to W$ and $P' \to P$ are weak equivalences. Can we construct $X'$ such that $X' \to X$ will be weak equivalence and $X'Y'Z'W'$, $X'Y'XY$, and $X'XZ'Z$ are homotopy commutative?
Thank you so much in advance. Any help will be appreciated.