The asymptotic lower-bound conjecture for edge-coloring components

Let $K_n^r$ be the complete $r$-uniform hypergraph, let $f(n,k,r)$ denote the minimum possible number of monochromatic components in an edge-coloring using at most $k$ colors, and let $z_{k,r}$ be the infimum, over all $n\geq r$, of the minimum possible maximum fraction of vertices incident with an edge of a single color in such a coloring.

Asymptotic lower-bound conjecture. For $k\geq r+1$,

$$f(n,k,r)=n\left(\frac{1}{r}-\frac{z_{k,r}}{r}\right)(1+o_n(1)).$$

The conjecture asserts that the lower bound in the preceding asymptotic theorem is asymptotically sharp. Infinitely many values of $z_{k,r}$ are known, while the values of infinitely many others remain open problems.