The q-zero forcing conjecture for the Cartesian product of a complete bipartite graph and K_2

Let $K_{n,m}$ be the complete bipartite graph with parts of sizes $n$ and $m$, and let $K_2$ be the complete graph on two vertices. Write $Z_q(G)$ for the $q$-analogue of the zero forcing number of a graph $G$. The q-zero forcing conjecture. For any $n,m\geq 2$,

$$Z_q(K_{n,m} \: \Box \: K_2)= \begin{cases} 2\min(n,m) & \text{if } q=0,\\ n+m & \text{for } q\geq 1. \end{cases}$$

This conjecture proposes exact values of the $q$-zero forcing number for this Cartesian-product family; the supplied text gives no resolution, so its status remains open.