Exactness conjecture for Hom-complex connectivity of $K_2\times K_n$

For a bipartite graph of the form $K_2\times K_n$, let $d$ denote its maximal degree and let $K_m$ be a complete graph. The lower bound for the connectivity of $\operatorname{Hom}(K_2\times K_n,K_m)$ given by Čukić and Kozlov is $m-d-2$. Exactness conjecture for $K_2\times K_n$. These lower bounds are exact for all bipartite graphs of the type $K_2\times K_n$. The paper has just established connectivity values for this family in several cases and presents this broader exactness assertion as a conjecture; no resolution is supplied in the given material.