Crawley-Boevey's conjectural solution to the multiplicative Deligne–Simpson problem

Let $\mathcal{C}=(C_j)_{1\le j\le k}$ be a tuple of conjugacy classes of $\operatorname{GL}_n$, defining a star-shaped quiver $Q$, a deformation parameter $\mathbf{q}$, and a dimension vector $\mathbf{d}$ as in the construction above. For the associated set $\Sigma_{\mathbf{q}}$ of roots, consider solutions $A_1\cdots A_k=1$ with $(A_j)_j\in\mathcal{C}$. Crawley-Boevey's conjecture. The following statements are equivalent: (i) there is an irreducible solution to $A_1\cdots A_k=1$ with $(A_j)_j\in\mathcal{C}$; (ii) $\mathbf{d}\in\Sigma_{\mathbf{q}}$. This conjecture gives a root-theoretic criterion for the existence of irreducible solutions to the multiplicative Deligne–Simpson problem. Its resolution status is not specified in the supplied source context.