Join conjecture with maximum multiplicity parameter

Let $G$ and $H$ be graphs such that $q(G)=q(H)=2$, with $MB(G)=k_1$ and $MB(H)=k_2$, and suppose that the quantities denoted in the source by $\mrr+(G)$ and $\mrr+(H)$ satisfy $\mrr+(G)=k_1$ and $\mrr+(H)=k_2$. Join maximum-multiplicity conjecture. Then $q(G\vee H)=2$ and $MB(G\vee H)=k$, where

$$k=\max\{k_1,k_2\}.$$

This is a proposed generalization of the preceding join inequality; the custom notation $\mrr+$ is not defined in the supplied context.