Polynomiality conjecture for indecomposable parabolic-bundle counts

Let $\mathbf{m}=(m_1,\dots,m_f)\in\mathbb{Z}_{\geq 0}^f$ satisfy $\sum_{i=1}^f m_i=n$. For $\mathbf{d}=(d_1,\dots,d_r)\in\mathbb{Z}_{\geq 0}^r$ and $\boldsymbol{\mu}=(\mu^1,\dots,\mu^r)\in(\mathcal{P}_n)^r$, let $\mathcal{A}_{\boldsymbol{\mu},D}^{\mathcal{E}}(q)$ count geometrically indecomposable parabolic structures of the prescribed types on a vector bundle $\mathcal{E}$ of type $\mathbf{m}$.

Polynomiality conjecture. There exists a polynomial $A_{\boldsymbol{\mu};\mathbf{d}}^{\mathbf{m}}(t)\in\mathbb{Z}[t]$ such that for every finite field $\mathbb{F}_q$ and every divisor $D=\sum_{i=1}^r\mathfrak{a}_i$ on $\mathbb{P}^1_{\mathbb{F}_q}$ with $\deg(\mathfrak{a}_i)=d_i$, one has

$$A_{\boldsymbol{\mu};\mathbf{d}}^{\mathbf{m}}(q)=\mathcal{A}_{\boldsymbol{\mu},D}^{\mathcal{E}}(q),$$

where $\mathcal{E}$ is a vector bundle of type $\mathbf{m}$.

This conjecture would make the counting function for geometrically indecomposable parabolic structures polynomial in $q$; the source derives it from the preceding nilpotent-count conjecture.