Strong Stanley conjecture for Jack Littlewood–Richardson coefficients

Let $c_{\mu\nu}^{\lambda}$ be the ordinary Littlewood–Richardson coefficient. For a triple of partitions $(\mu,\nu,\lambda)$, let $\mathsf{SD}_{\mu\nu}^{\lambda}$ denote the set of Stanley diagrams, and let $[\bm D]$ denote evaluation of the corresponding hook-symbol rational function at the associated $\alpha$-hook lengths. Strong Stanley conjecture. If

$$c_{\mu\nu}^{\lambda}=1,$$

then there exists a Stanley diagram $\bm D\in\mathsf{SD}_{\mu\nu}^{\lambda}$ such that

$$g_{\mu\nu}^{\lambda}=[\bm D].$$

Moreover, $\bm D$ may be chosen with equal numbers of upper hooks in its numerator and denominator, and hence equal numbers of lower hooks. This is Stanley’s motivating conjecture for the diagrammatic formulation; the source does not state a resolution.