The supersaturation conjecture for independent transversals

Let $G$ be a graph with maximum degree $\Delta$ and let $\mathcal{U}$ be a $(2\Delta)$-thick partition of $V(G)$. An independent transversal is a set containing exactly one vertex from each part of $\mathcal{U}$ and inducing no edges in $G$. Supersaturation conjecture. The number of independent transversals of $(G,\mathcal{U})$ is minimised when $G$ is a disjoint union of $|\mathcal{U}|$ copies of $K_{\Delta,\Delta}$. This conjecture proposes a sharp supersaturation-type strengthening of Haxell's theorem; the supplied text gives no resolution, so its status remains open.