Multipartite form of the generalized Ohba conjecture

Let $r\ge 2$, and let $p_1,p_2,\ldots,p_k$ be positive integers. For each $i$, let $V_i$ be a part of size $p_i$, and let $K_{p_1,p_2,\ldots,p_k}^r$ be the $r$-complete $k$-partite hypergraph whose edges are the $r$-subsets not contained in a single part. Assume that $p_i\ge r-1$ for all $i\in\{1,2,\ldots,k\}$, with at most one exception. Multipartite generalized Ohba conjecture. If

$$\sum\limits_{i=1}^{k} p_i\le rk+r-1,$$

then

$$\chi_l\left(K_{p_1,p_2,\ldots,p_k}^r\right)=k.$$

The paper states that the generalized Ohba conjecture is equivalent to its validity for all $r$-complete multipartite hypergraphs and uses the displayed proposition to obtain this formulation. It remains open in the supplied source.