The cellular tree normal form conjecture for quiver roots

Let $Q$ be a quiver and let $\alpha$ be a root for $Q$. Write $c_i$ for the coefficients in the polynomial $a_\alpha(q)=\sum_i c_iq^i$ counting absolutely indecomposable representations over finite fields. A cellular tree normal form means a cellular normal form for the indecomposable representations in which the representatives are tree modules.

Cellular tree normal form conjecture. The root $\alpha$ admits a cellular tree normal form, with $c_i$ cells of dimension $i$.

This conjecture refines Kac's cellular decomposition conjecture by proposing tree modules as a mechanism for constructing the cells. The paper presents it as likely difficult and does not state that it has been resolved.