The window positivity conjecture for Jack Littlewood–Richardson coefficients

Let $\mu\subset\lambda$ satisfy $\lambda/\nu\subset R$, where $R$ is a window. Let $F_{\mu,R}^{\lambda}$ be the window factor and define

$$j_{\lambda,R}=\prod_{b\in\lambda\cap R}h^U_\lambda(b)h^L_\lambda(b).$$

Window positivity conjecture. The window factor accounts for all boxes outside $R$ in the positivity expression, namely

$$g_{\mu\nu}^{\lambda}\left(F_{\mu,R}^{\lambda}\right)^{-1}j_{\lambda,R}\in\mathbb Z_{\geq0}[\alpha].$$

This sharpens Stanley's positivity conjecture by localizing the contribution of the window complement. It is proposed conjecturally and supported by examples in the paper.