The coprime-width formula for the generalized descent difference polynomial

Let $n$ and $k$ be positive integers with $1\leq k<n$, and define

$$G_{n,k}(q)=\sum_{\sigma\in\mathfrak{S}_n}q^{\operatorname{des}_k(\sigma)-\operatorname{des}_{n-k}(\sigma)}.$$

Here $A_{n-1}(q)$ denotes the Eulerian polynomial of degree $n-1$. Coprime-width formula. If $\gcd(k,n)=1$, then

$$G_{n,k}(q)=nq^{1-k}A_{n-1}(q).$$

The formula was supported by computational data for all $n\leq 9$ and all $1\leq k<n$ with $\gcd(k,n)=1$, but it does not hold more generally; determining whether a general formula exists remains open.