Counterexamples to Clique Immersion Conjecture for Direct Products

Let $G$ and $H$ be graphs, and let $G\times H$ denote their direct product. For a graph $G$, let $\mathrm{im}(G)$ be the largest integer $t$ such that $G$ contains a $K_t$-immersion. Collins, Heenehan, and McDonald conjectured that if $\mathrm{im}(G)=t$ and $\mathrm{im}(H)=r$, then $$\mathrm{im}(G\times H)\ge (t-1)(r-1)+1.$$ We disprove this conjecture by constructing an infinite family of connected bipartite counterexamples.