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

Let $\bar{C}(n,k)$ denote the relevant Poincaré series, and define

$$F(t)=\frac{\prod\{1-t^{|i-j|}\mid 1\le i,j\le n,\ i\ne j\}}{\prod\{(1-t^{|i-j+1|})^k\mid 1\le i,j\le n,\ i\ne j-1\}}.$$

The function $F(t)$ can be written as $\prod_{i=1}^n(1-t^i)^{-\alpha(i)}$, where

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

Least-denominator conjecture. $\bar{C}(n,k)$ is a rational function with least denominator equal to the denominator of the $(n-1)(k-1)$-st derivative of $F(t)$.

The claim proposes a general denominator formula extending the computations for small values of $n$ and $k$, where the displayed denominator appeared to be least. The source does not provide a proof or resolution of the conjecture.