Carnevale–Voll's factorisation conjecture for multiset Euler–Mahonian polynomials

Let $\eta$ be a composition, and let $S_{\eta}$ denote the associated multiset permutations. For a polynomial $f\in\mathbb{Z}[x,y]$, call it unitary if there are $F\in\mathbb{Z}[Y]$ and $a,b\in\mathbb{N}_0$ such that $f(x,y)=F(x^ay^b)$ and every complex root of $F$ has absolute value $1$.

Carnevale–Voll's factorisation conjecture. The polynomial giving the joint distribution of $(\operatorname{den},\operatorname{exc})$ over $S_{\eta}$ has a unitary factor if and only if $\eta=(m^r)$ is a rectangle with $r$ even and $m$ odd. In that case,

$$\sum_{w\in S_{\eta}}x^{\operatorname{den}(w)}y^{\operatorname{exc}(w)}=\left(1+x^{\frac{rm}{2}}y\right)f^{\eta}_0(x,y),$$

where $f^{\eta}_0(x,y)$ has no unitary factor.

This is a reformulation of the factorisation conjecture for joint $(\operatorname{des},\operatorname{maj})$ distributions over multiset permutations. The stated factorisation singles out the unitary factor $1+x^{rm/2}y$ in exactly the even-by-odd rectangular cases; its resolution is not given in the supplied text.