Least-denominator conjecture for the Poincaré series of generic matrix representations

Let $G(t)$ be the rational function

$$G(t)=\frac{(\sum_{i,j=1}^n t^i)^2\prod\{1-t^{i-j}\mid 1\le i,j\le n,\ i\ne j\}}{\prod\{(1-t^{i-j+1})^k\mid 1\le i,j\le n,\ i\ne j-1\}},$$

whose extra numerator factor is $(1+t+\cdots+t^{n-1})^2$. Equivalently, if $G(t)=\prod_i(1-t^i)^{-\beta(i)}$, then

$$\beta(i)=\begin{cases}2(k-1)(n-1)+2,&i=1,\\ 2(k-1)(n-i),&2\le i\le n-1,\\ k-2,&i=n.\end{cases}$$

Here $\bar{R}(n,k)$ denotes the corresponding Poincaré series. Least-denominator conjecture for $\bar{R}(n,k)$. The function $\bar{R}(n,k)$ is rational, and its least denominator is the same as the denominator of the $(n-1)(k-1)$-st derivative of $G(t)$. The conjecture is supported by the agreement of the analogous denominator computations in the cases discussed in the paper, but the general assertion remains unproved.