A join upper-bound conjecture for the hyperopic cop number

Let $G$ and $J$ be connected graphs. Write $G\vee J$ for their graph join, $c_H(X)$ for the hyperopic cop number of a graph $X$, and let $\Upsilon(X)$ denote the maximum size of a common neighbourhood set in $X$. Join upper-bound conjecture.

$$c_H(G\vee J) \leq \min\{c_H(G),\Upsilon(G)\}+\min\{c_H(J),\Upsilon(J)\}.$$

This is posed as an open question concerning small common neighbourhood sets. The difficulty is that the robber can access any vertex in the other graph, potentially interfering with winning strategies played by cops within $G$ and $J$; the paper also records weaker upper bounds involving $c(G\vee J)$ and the common-neighbourhood parameters.