Stanley's strong conjecture for Jack Littlewood–Richardson coefficients when the Schur coefficient is one

Let $g_{\mu\nu}^\lambda$ be the Jack Littlewood–Richardson coefficient and let $c_{\mu\nu}^\lambda$ be the corresponding Schur Littlewood–Richardson coefficient. For every box $b$, let $h^U_\sigma(b)$ and $h^L_\sigma(b)$ denote the upper and lower hook symbols associated with a partition $\sigma$.

Strong Stanley conjecture. If $c_{\mu\nu}^\lambda=1$, then

$$g_{\mu\nu}^\lambda=\frac{\displaystyle\prod_{b\in\mu}h^{\Phi_b}_\mu(b)\,\prod_{b\in\nu}h^{\Phi_b}_\nu(b)}{\displaystyle\prod_{b\in\lambda}h^{\Phi_b}_\lambda(b)},$$

where each $\Phi_b$ is either $U$ or $L$, and the symbols $h^L$ and $h^U$ occur equally often in the numerator and denominator. This is a proposed explicit, manifestly positive form of Stanley's general conjecture in the case $c_{\mu\nu}^\lambda=1$.