Conjectured equivalence for commuting-pair counts of quiver endomorphism algebras

Let $Q$ be an acyclic quiver and let $\mathbf{d}\in\mathbb{Z}_{\geq 0}^{Q_0}$. For each $\mathbf{e}\leq\mathbf{d}$, let $[Q,\mathbf{e}]$ denote the relevant commuting-pair count in the endomorphism algebra associated with the projective representation of summand vector $\mathbf{e}$, and let $\overline{[Q,\mathbf{e}]}$ denote the number of commuting pairs in $A_{Q,\mathbf{e}}\times\operatorname{rad}A_{Q,\mathbf{e}}$. Commuting-pair equivalence conjecture. The quantity $\overline{[Q,\mathbf{e}]}$ is a polynomial in $q$ for all $\mathbf{e}\leq\mathbf{d}$ if and only if $[Q,\mathbf{e}]$ is a polynomial in $q$ for all $\mathbf{e}\leq\mathbf{d}$. Moreover, $\overline{[Q,\mathbf{d}]}$ can be expressed in terms of $[Q,\mathbf{e}]$ for $\mathbf{e}\leq\mathbf{d}$. The paper presents this as an analogue of a classical result for commuting pairs in $B_n\times T_n$ and as a question related to the quiver version of Higman's conjecture.