Strong weighting conjecture for nice uniform hypergraphs

Let $H$ be a nice $r$-uniform hypergraph, where $r\geq 3$. A weight function is a map $\omega:E\to\{1,\dots,w\}$, inducing the vertex-coloring $c:V\to\mathbb{N}$ by

$$c(v)=\sum_{e\ni v}\omega(e).$$

The coloring is strong if the vertices in every edge receive pairwise distinct colors, and $H$ is strongly $w$-weighted if some such weight function induces a strong coloring.

Strong weighting conjecture. For every $r\geq 3$, there is a constant $w=w(r)$ such that each nice $r$-uniform hypergraph is strongly $w$-weighted.

The conjecture proposes a general upper bound for strong weightings, after the paper's NP-completeness result for deciding strong 2-weightedness. The source further notes lower bounds on any such $w$ for infinitely many and for all values of $r$, but does not resolve the conjecture.