Hodges's strengthened conjecture on tangled labelings

Let $P$ be a poset with $n$ elements and $m$ minimal elements. A labeling of $P$ is tangled if it requires $n-1$ applications of extended promotion to become a natural labeling. Hodges's conjecture. The number of tangled labelings of $P$ is at most

$$(n-m)(n-2)!.$$

Hodges's bound strengthens the Defant–Kravitz bound and is known in the supplied context for inflated rooted forest posets; its general validity remains open.