Large-degree conjecture for multipartite hypergraphs with a heavy edge

Let $r\geq 4$ and let $H$ be an $r$-partite $r$-uniform multi-hypergraph with $n$ edges. Let $\nu:=\nu_{2}(H)$ denote its matching parameter.

Large-degree conjecture. There exists $d_r>0$ such that if $H$ has an edge of multiplicity at least $\nu+1$, then

$$\Delta(H)\geq \frac{n}{r-1}-d_r\nu.$$

This statement is proposed as the generalization needed to extend the preceding results to the case $k=2$ and $r\geq 4$. The source presents it as an unresolved sufficient step for the corresponding problem.