Generalized covering conjecture for complete $(r,\ell)$-partite hypergraphs

Let $r\geq 3$, $1\leq \ell\leq r$, and $k\geq 1+r-\ell$. A complete $r$-uniform $(r,\ell)$-partite hypergraph has its vertices partitioned into $r$ nonempty classes, and contains every $r$-set meeting each class in at most $\ell$ vertices. For a spanning $k$-coloring, let $\mathrm{cov}(r,\ell,k)$ be the maximum, over such colorings, of the minimum number of monochromatic connected components needed to cover the vertex set.

Generalized covering conjecture.

$$\mathrm{cov}(r,\ell,k)=1+\Bigl\lfloor \frac{k-r+\ell-1}{\ell}\Bigr\rfloor$$

for every $r\geq 3$, $k\geq 1+r-\ell$, and $1\leq \ell\leq r$.

The paper proves the formula except when $\ell=1$ and $k\geq 2r$, where only bounds differing by one are established. Thus the unresolved cases coincide with the missing cases of the complete $r$-partite conjecture.