Signed cycle-type restriction polynomiality conjecture

Let $n$ be a positive integer, let $k$ be a fixed positive integer with $1\leq k\leq n$, and let $\mathfrak{S}_n$ denote the symmetric group on $n$ elements. For $\pi\in\mathfrak{S}_n$, let $\operatorname{type}(\pi)$ be its cycle type, and let $\operatorname{RLMv}(\pi)$ and $\operatorname{EXCv}(\pi)$ denote its right-to-left-minimum-value and excedance-value multisets, respectively. Then the polynomiality conjecture.

$$\sum_{\substack{\pi\in\mathfrak{S}_n\min(\operatorname{type}(\pi))\geq k}}(-1)^{\operatorname{inv}(\pi)}\mathbf{x}_{\operatorname{RLMv}(\pi)}\mathbf{y}_{\operatorname{EXCv}(\pi)}$$

is an element of $\mathbb{N}[x_1,\dotsc,x_n,y_1,\dotsc,y_n]$. This generalizes the derangement case by imposing a lower bound on the lengths of all cycles; the source provides no resolution or further evidence for the claim, so its status remains open.