Quiver analogue of Higman's conjecture for commuting radical endomorphisms

Let $Q$ be an acyclic quiver, let $\mathbb{k}=\mathbb{F}_q$, and let $P\in\operatorname{rep}Q$ be a projective representation over $\mathbb{k}$. Consider the radical $\operatorname{rad}(\operatorname{End}(P))$ of its endomorphism algebra. Quiver Higman conjecture. The number of commuting pairs in $\operatorname{rad}(\operatorname{End}(P))$ is a polynomial in $q$. The paper introduces this as a quiver generalization of Higman's conjecture and states in the abstract that it is solved for quivers containing no path of length exceeding two, while the general case remains open.