The counting conjecture for joins of critical graphs and cliques

Let $G$ be a $k$-critical graph. For a graph $F$, write $P(F,r)$ for the number of proper $r$-colorings of $F$ using the colors $1,\ldots,r$, and let a $r$-fold cover of $F$ mean a cover in the DP-coloring sense with parameter $r$.

Counting conjecture. For all sufficiently large $t$ and every $(k+t)$-fold cover $\mathcal{H}$ of $G\vee K_t$, the number of proper $\mathcal{H}$-colorings of $G\vee K_t$ is at least

$$P(G\vee K_t,k+t).$$

This is presented as an open special case of a broader enumerative question for robustly critical graphs: whether every $k$-fold cover has at least as many proper colorings as the chromatic polynomial predicts. The source states that this special case is open, although analogous assertions are known for complete graphs, odd cycles, and joins of odd cycles with cliques.