The weighted clique-packing conjecture for uniform weights on a clique independent set

Let $G$ be a graph, let $m\geq 1$ and $k\geq 2$, and let $w$ be an $m$-weighting of $G$. A $k$-clique independent set is a set $I$ of vertices such that every $k$-clique of $G$ contains exactly one vertex of $I$. An $m$-weighting $w'$ is uniform on $I$ when the weights assigned by $w'$ to the vertices of $I$ are equal. Weighted clique-packing conjecture. There exist a $k$-clique independent set $I$ of $G$ and an $m$-weighting $w'$ of $G$ uniform on $I$ such that

$$\pi_k(G(w'))\leq\pi_k(G(w)).$$

for every $m$-weighting $w$ of $G$. The preceding theorems establish the analogous minimization statement for several graph classes, while the theorem for $k=3$ shows that uniformity on all vertices need not attain the minimum; the conjecture proposes that uniformity on a suitable $k$-clique independent set always suffices.