The extension-weight conjecture for strict partitions

Let $\tilde{\boldsymbol{ối}}$ be a strict partition, let $n\in\mathbb{N}$, and let $\sigma\in\operatorname{Tab}(\tilde{\boldsymbol{\bối}}^{\mathrm{c}},n)$ be a filling. Let $\operatorname{Ext}_{\tilde{\boldsymbol{b}}}(\sigma)$ be the set of fillings of the diagram of $\tilde{\boldsymbol{b}}$ extending $\sigma$, and let $\operatorname{wt}$ denote their stationary weights. Then Extension-weight conjecture.

$$\frac{1}{\operatorname{wt}(\sigma)}\sum_{T\in\operatorname{Ext}_{\tilde{\boldsymbol{b}}}(\sigma)}\operatorname{wt}(T)=\frac{\widetilde{H}_{\tilde{\boldsymbol{b}}}(x_1,\dots,x_n;1,t)}{\widetilde{H}_{\tilde{\boldsymbol{b}}^{\mathrm{c}}}(x_1,\dots,x_n;1,t)}.$$

This would provide a combinatorial proof of the reduced partition-function result for strict partitions by showing that the extension factor is independent of $\sigma$. The paper presents it as a refined conjecture and supplies no proof.