Partitioned Kneser hypergraph chromatic-number conjecture

Let $r\ge 2$, $k\ge 1$, and $n\ge rk$ be integers, and let $\mathcal P=\{P_1,\dots,P_\ell\}$ be a partition of $[n]$ such that $|P_i|\le r$ for every $i$. Let $\mathrm{KG}^r(n,k;\mathcal P)$ denote the corresponding partitioned $r$-uniform Kneser hypergraph. Partitioned Kneser conjecture. Then

$$\chi\bigl(\mathrm{KG}^r(n,k;\mathcal P)\bigr)=\left\lceil\frac{n-r(k-1)}{r-1}\right\rceil.$$

The source explains that this would remove the power-of-two restriction in the preceding lemma and would imply the generalized Erdős–Kneser conjecture in full generality; it remains open in the supplied text.