Is the condition in Corollary 5.43 also sufficient? In other words, let $N$, $\Sigma$ and $\Psi$ be as before and let $\psi$ be a rational piecewise affine function on $N_{\mathbb{R}}$ such that $\psi-\Psi$ can be extended to a continuous function on $N_{\Sigma}$. Does it exists a complete SCR polyhedral complex $\Pi$ with $\operatorname{rec}(\Pi)=\Sigma$ and $\psi$ is piecewise affine on $\Pi$?