Kamat's cross-intersection conjecture for independent sets of graphs

Let $G$ be a graph, let $\mathcal{I}_G$ be the family of independent sets of $G$, and let $\mathcal{I}_G^{(r)}$ be its $r$-th level. Two subfamilies are cross-intersecting if every member of one intersects every member of the other. Write $\mu(\mathcal{I}_G)$ for the minimum base size.

Kamat's conjecture. If $\mu(\mathcal{I}_G)\geq 2r$ and $\mathcal{A},\mathcal{B}\subseteq\mathcal{I}_G^{(r)}$ are cross-intersecting, then

$$|\mathcal{A}|+|\mathcal{B}|\leq |\mathcal{I}_G^{(r)}|.$$

The source presents this as an analogue of the Holroyd–Talbot conjecture and does not state that it has been resolved.