Bosek's asymptotic harmonious coloring conjecture for hypergraphs

Let $k,\Delta\geq 2$. A harmonious coloring of a $k$-uniform hypergraph $H$ is a vertex coloring that is rainbow on every edge and assigns distinct color sets to distinct edges; let $h(H)$ be the least number of colors in such a coloring. If $H$ has $m$ edges and maximum degree $\Delta$, then Bosek's conjecture. There exists a constant $c=c(k,\Delta)$ such that

$$h(H)\leq \sqrt[k]{k!m}+c.$$

The lower bound $\binom{r}{k}\leq m$ for a harmonious coloring using $r$ colors gives the leading-order term $\sqrt[k]{k!m}$, so the conjecture asks whether the known upper bound can be sharpened to this term up to an additive constant depending only on $k$ and $\Delta$.