The nonnegative hook-content formula conjecture for skew partitions

Let $\overline{H}_{\lambda/\mu}(n)$ be the normalized hook-content quantity associated with partitions $\mu\subseteq\lambda$, and let $\ell$ denote the length of $\lambda$. For an integer $n\geq\ell$, the nonnegative hook-content formula conjecture.

$$\overline{H}_{\lambda/\mu}(n)\in\mathbb{Z}_{\geq 0}.$$

The claim asserts integrality and nonnegativity of the formula, although these properties are not evident from its definition. The authors report verification in numerous cases; no general proof or combinatorial interpretation is supplied.