Integer-polynomial positivity conjecture for stable-center structure constants

Let $\Phi_q$ be the set of monic irreducible polynomials over $\mathbb F_q$ other than $t$, and let $\Phi_{\mathbb Z}$ be the set of monic irreducible polynomials in $\mathbb Z[t]$ other than $t$. For $\boldsymbol{\lambda},\boldsymbol{\mu},\boldsymbol{\nu}\in\mathcal P(\Phi_{\mathbb Z})$, view their supports over $\mathbb F_q$ whenever $\Phi_{\mathbb Z}(\boldsymbol{\lambda}),\Phi_{\mathbb Z}(\boldsymbol{\mu}),\Phi_{\mathbb Z}(\boldsymbol{\nu})\subset\Phi_q$.

Integer-polynomial positivity conjecture. There is a polynomial $A_{{\boldsymbol{\lambda}}{\boldsymbol{\mu}}}^{{\boldsymbol{\nu}}}({\bf q})\in\mathbb Z[{\bf q}]$ such that

$$a^{{\boldsymbol{\nu}}}_{{\boldsymbol{\lambda}}{\boldsymbol{\mu}}}=A_{{\boldsymbol{\lambda}}{\boldsymbol{\mu}}}^{{\boldsymbol{\nu}}}(q)$$

for every prime power $q$ satisfying the support condition. Define $\widetilde A_{{\boldsymbol{\lambda}}{\boldsymbol{\mu}}}^{{\boldsymbol{\nu}}}({\bf q})=A_{{\boldsymbol{\lambda}}{\boldsymbol{\mu}}}^{{\boldsymbol{\nu}}}({\bf q}+1)$. Then

$$a^{{\boldsymbol{\nu}}}_{{\boldsymbol{\lambda}}{\boldsymbol{\mu}}}=\widetilde A_{{\boldsymbol{\lambda}}{\boldsymbol{\mu}}}^{{\boldsymbol{\nu}}}(q-1),$$

and $\widetilde A_{{\boldsymbol{\lambda}}{\boldsymbol{\mu}}}^{{\boldsymbol{\nu}}}\in\mathbb N[{\bf q}]$.

This conjecture proposes polynomial dependence on $q$ together with positivity after shifting the variable by one. It is supported by all examples computed in the cited section, but no proof or resolution is given.