Prime-monomial quiver Grassmannian coefficient conjecture

Let $Q$ be a quiver and let $v_1,\ldots,v_n$ be a sequence of mutations. For each vertex $i$, let $\phi_{n,i}$ be the representation on the base quiver corresponding to the last mutation up to and including the $n$th step that occurred at vertex $i$. Call a monomial $p$ prime if every subrepresentation of $\phi_{n,i}$ with dimension vector equal to the degree vector of $p$ is indecomposable. Prime-monomial coefficient conjecture. The space of such subrepresentations has Euler–Poincaré characteristic equal to the $i$th component of $-C_n^{-1}(p)$, namely

$$-C_n^{-1}(p)_i.$$

This conjecture is intended to identify coefficients of deformed $F$-polynomials with Euler characteristics of quiver Grassmannians; the source gives no resolution.