By the proof of Theorem 4.97 and Corollary 5.43, when $\psi$ is concave, the conditions

1. (1)

   $|\psi-\Psi|$ is bounded;

2. (2)

   $\psi-\Psi$ can be extended to a continuous function on $N_{\Sigma}$;

3. (3)

   there exist a complete SCR polyhedral complex $\Pi$ with $\operatorname{rec}(\Pi)=\Sigma$ and $\psi$ piecewise affine on $\Pi$;

are equivalent. In particular, the answer to the above question is positive when $\psi$ is concave.