Hook-correspondence conjecture for adjacent Jack coefficients

Let $(\mu_1,\nu_1,\lambda_1)$ and $(\mu_2,\nu_2,\lambda_2)$ be adjacent triples of partitions corresponding to a pivot at $b$. Let $\mathcal W'$ and $\mathcal W$ be the associated window families, and let $\mathsf R$ denote the corresponding spaces of rules modulo hook symbols. Hook-correspondence conjecture. There exists a family of hook correspondences $\widetilde\psi$ inducing a map

$$\{\widetilde\psi\}:\mathsf R_{\mu_1\nu_1}^{\lambda_1}(\mathcal W')/\{\bm h_b^{\mathrm U}\}\longrightarrow\mathsf R_{\mu_2\nu_2}^{\lambda_2}(\mathcal W)/\{\bm h_{b'}^{\mathrm L}\},$$

under which

$$\widetilde\psi:\bm g_{\mu_1\nu_1;\lambda_1}\bmod\bm h_{\sigma_1}^{\mathrm A}(b)\longmapsto\bm g_{\mu_2\nu_2;\lambda_2}\bmod\bm h_{\sigma_2}^{\mathrm A}(b).$$

This generalizes the displayed local equivalence between the two example triples and is intended to explain the adjacent-coefficient congruences. The source does not report a proof.