Adjacent-triple congruence conjecture for Jack Littlewood–Richardson coefficients

Let $(\mu_1,\nu_1,\lambda_1)$ and $(\mu_2,\nu_2,\lambda_2)$ be triples of partitions that differ by a single box move in one pair, with pivot boxes $b_1\in\sigma_1$ and $b_2\in\sigma_2$ satisfying

$$h_{\sigma_1}^{\mathrm U}(b_1)=h_{\sigma_2}^{\mathrm L}(b_2).$$

Here $h_{\sigma}^{\mathrm U}(b)$ and $h_{\sigma}^{\mathrm L}(b)$ denote the upper and lower $\alpha$-hook lengths, respectively. Adjacent-triple congruence conjecture. The corresponding Jack Littlewood–Richardson coefficients satisfy

$$g_{\mu_1\nu_1;\lambda_1}(\alpha)\equiv g_{\mu_2\nu_2;\lambda_2}(\alpha)\pmod {h_{\sigma_1}^{\mathrm U}(b_1)}.$$

This gives the precise hook-divisibility relation suggested in the abstract and is intended as a compatibility relation between adjacent triples. The source presents it as a conjecture and gives no resolution evidence.