Kostov's irreducible Deligne–Simpson criterion

Let $C_1,\ldots,C_k$ be the prescribed conjugacy classes, and let $A_i\in C_i$. Let $\alpha$ be the associated dimension vector, let $\xi^{[\delta]}$ denote the scalar attached to a positive root $\delta$, and let $p(\delta)$ be the corresponding function on dimension vectors. An irreducible solution means a solution whose matrices $A_i$ have no common invariant subspace.

Kostov's conjecture. There is an irreducible solution to

$$A_1\dots A_k=1$$

with $A_i\in C_i$ if and only if $\alpha$ is a positive root, $\xi^{[\alpha]}=1$, and

$$p(\alpha)>p(\beta)+p(\gamma)+\dots$$

for every nontrivial decomposition $\alpha=\beta+\gamma+\dots$ into positive roots satisfying $\xi^{[\beta]}=\xi^{[\gamma]}=\dots=1$.

This is the irreducible form of the Deligne–Simpson problem, which asks for matrices in prescribed conjugacy classes whose product is the identity. The criterion is presented here as a conjecture; the supplied source gives no resolution evidence.