A linear lower bound for directed-path length in terms of girth and minimum out-degree

For a digraph $D$, let $ell(D)$ be the length of a longest directed path, let $g(D)$ be its girth, and let $\delta^+(D)$ be its minimum out-degree. Proposed weaker Thomassé conjecture. There is some constant $c>0$ such that

$$\ell(D) \ge c\,g(D)\,\delta^+(D)$$

for every digraph $D$.

This conjecture weakens Thomassé's proposed bound and is motivated by the paper's lower bounds for regular digraphs. The supplied text does not state whether it is known or remains open, so its database status is left open.