Jack windowing conjecture

Let $T$ be a triple of partitions and let $W$ be a window for $T$. A rule is a Stanley sum whose evaluation gives the corresponding Jack Littlewood–Richardson coefficient. Jack windowing conjecture. There exists a rule $\bm g_T$ such that a rule $\bm g_W$ can be constructed by assigning a specified hook choice $\bm F_W$ to boxes outside $T\subset W$, assigning the same rule $\bm g_T$, extended by linearity, to boxes inside $T$, and satisfying

$$\bm g_W=\bm F_W\circ(\bm g_T)_W.$$

This conjectures that rules for root triples extend uniformly across their window families. The source provides the window definitions but no proof or resolution.