Stahl's conjecture on multichromatic numbers of Kneser graphs

Let $k$, $d$, and $s$ be integers with $d\geq 2s$. The $k$th multichromatic number $\chi_k(G)$ is the smallest number of colors needed to assign each vertex of a graph $G$ a set of $k$ colors so that adjacent vertices receive disjoint sets. The Kneser graph $K(d,s)$ has as vertices the $s$-subsets of $[d]=\{1,\ldots,d\}$, with two vertices adjacent when the corresponding sets are disjoint.

Stahl's conjecture. For all integers $k$ and $d\geq 2s$,

$$\chi_k(K(d,s))=\left\lceil\frac{k}{s}\right\rceil\cdot(d-2s)+2k.$$

This conjecture concerns the multichromatic numbers of Kneser graphs and has applications to hardness results for graph coloring and orthogonality dimension. More than forty years after it was proposed, it remains open.