Witten's conjecture for open $r$-spin intersection numbers in higher genus

Let $r\ge 2$. For $g\ge 1$, denote by $\langle\tau^{\alpha_1}_{d_1}\cdots\tau^{\alpha_l}_{d_l}\sigma^k\rangle^{\frac{1}{r},o}_g$ the proposed open $r$-spin intersection numbers, and define

$$F_g^{\frac{1}{r},o}(t^*_*,s):=\sum_{l,k\ge 0}\frac{1}{l!k!}\sum_{\substack{0\le\alpha_1,\ldots,\alpha_l\le r-1\\\d_1,\ldots,d_l\ge 0}}\langle\tau^{\alpha_1}_{d_1}\cdots\tau^{\alpha_l}_{d_l}\sigma^k\rangle^{\frac{1}{r},o}_g t^{\alpha_1}_{d_1}\cdots t^{\alpha_l}_{d_l}s^k.$$

Here $t^a_d$ and $s$ are formal variables, and $\phi_g$ denotes the genus-$g$ contribution to the extended closed $r$-spin theory. Open higher-genus $r$-spin conjecture. For every $g\ge 1$, there is a geometric construction of these open $r$-spin intersection numbers, and their generating series satisfies

$$F^{\frac{1}{r},o}_g=\left.(-r)^{\frac{g-1}{2}}\phi_g\right|_{t^{r-1}_d\mapsto\frac{1}{\sqrt{-r}}(t^{r-1}_d-\delta_{d,0}rs)}.$$

This predicts a higher-genus extension of the genus-zero open $r$-spin theory and relates it to the extended closed theory; the paper presents geometric and algebraic evidence but does not establish the construction or identity.