Exact two-Schur correlator formula for the r-fold q-deformed logarithmic matrix model

Let $N,r$ be positive integers, let $q$, $\Lambda$, $u$, $v$, and $a$ be the model parameters, and let $\omega_r$ denote an $r$th root of unity. For partitions $\lambda$ and $\mu$, define $Z_{\lambda\mu}$ by

$$Z_{\lambda\mu}=\prod_{1\leq i<j\leq N}\frac{\lfloor\lambda_i-\lambda_j+j-i;\,0\rfloor_{r,0}}{\lfloor j-i;\,0\rfloor_{r,0}}\prod_{1\leq i<j\leq N}\frac{\lfloor\mu_i-\mu_j+j-i;\,0\rfloor_{r,0}}{\lfloor j-i;\,0\rfloor_{r,0}}$$ $$\mathrel{\phantom{Z_{\lambda\mu}=}}\times\prod_{1\leq i,j\leq N}\frac{\lfloor2N+1-i-j;\,u+v\rfloor_{r,a}}{\lfloor2N+1+\lambda_i+\mu_j-i-j;\,u+v\rfloor_{r,a}} \prod_{(i,j)\in\lambda}\frac{\lfloor N+j-i;\,u\rfloor_{r,a}}{\lfloor N+j-i;\,u+v\rfloor_{r,a}} \prod_{(i,j)\in\mu}\frac{\lfloor N+j-i;\,v\rfloor_{r,0}}{\lfloor N+j-i;\,0\rfloor_{r,0}},$$

where

$$\lfloor x; y\rfloor_{r,a}=\begin{cases} \Lambda\,\dfrac{q^{(x+ry)/2}-q^{-(x+ry)/2}}{q^{1/2}-q^{-1/2}}, &(x+a)\bmod r=0,\\[6pt] \dfrac{\omega_r^{x/2}-\omega_r^{-x/2}}{\omega_r^{1/2}-\omega_r^{-1/2}}, &(x+a)\bmod r\neq0. \end{cases}$$

The exact two-Schur correlator conjecture. The average of the two Schur-polynomial insertions satisfies

$$\left\langle\chi_\lambda(X)\,\chi_\mu\left(\operatorname{Tr}X^k\mapsto\operatorname{Tr}X^k+rv\,\delta_{r\mid k}\right)\right\rangle= \begin{cases} Z_{\lambda\mu},&\text{if }\deg_\Lambda Z_{\lambda\mu}=0,\\ 0,&\text{otherwise.} \end{cases}$$

This formula proposes an exact expression for two-Schur correlators in the general $r$-fold $q$-deformed logarithmic model. The authors report testing it with numerous computer experiments, but the supplied text does not establish a proof or give evidence that the conjecture has been resolved.