Refined bound for tangled labelings of connected posets

Let $P$ be a connected $n$-element poset with $s$ minimal elements. A labeling of $P$ is tangled if its sorting time is $n-1$.

Refined tangled-labeling bound. $P$ has at most

$$(n-s)(n-2)!$$

tangled labelings. This is presented as a refinement of the bound attributed to Defant and Kravitz and is open in the stated generality.