Let $G=(V,E)$ be a finite simple graph. We say a map $p:V\to [n]:=\{1,\ldots,n\}$ is a pseudo-coloring if for all $a\neq b\in[n]$ there is $v\in\psi^{-1}(\{a\})$ and $w\in\psi^{-1}(\{b\})$ such that $\{v,w\}\in E$. We denote the maximal number $m$ such that there is a pseudo-coloring $p:V\to [m]$ by $\psi(V)$.

An easy argument shows that every coloring in the traditional sense is also a pseudo-coloring, which implies that $\psi(G) \geq \chi(G)$.

Let $G, H$ be finite simple undirected graphs. Do we necessarily have $\psi(G\times H) \leq \min\{\psi(G),\psi(H)\}$?

(By $G\times H$ we denote the categorical product, sometimes also referred to as the tensor product of graphs.)