Nonderogatory conjecture for quantum cohomology of Hilbert schemes of type-$A_r$ surfaces

Let $\mathcal{A}_r$ be the type-$A_r$ surface, let $\operatorname{Hilb}^n(\mathcal{A}_r)$ be its Hilbert scheme of $n$ points, and let $L$ be the map defined in Section 5. Consider the commuting family of operators

$$L((2))*_{\mathrm{crep}}-,\quad L(D_1)*_{\mathrm{crep}}-,\quad \ldots,\quad L(D_r)*_{\mathrm{crep}}-$$

on the quantum cohomology of $\operatorname{Hilb}^n(\mathcal{A}_r)$. Nonderogatory conjecture. This commuting family is nonderogatory; that is, its joint eigenspaces are one-dimensional. The conjecture concerns the simultaneous spectral structure of the crepant-resolution quantum cohomology operators and would imply strong control of the ring structure. The source gives no resolution, while noting that the statement is supported by partial evidence.