Generation conjecture for divisor multiplication operators on the Fock space

Let $\mathcal{F}_{\mathcal{A}_n}\cong \mathcal{F}_{[\mathbb{C}^2/\mathbb{Z}_{n+1}]}$ be the common Fock space, and let $D_0,\ldots,D_n$ be a basis of divisors. For each divisor, let $M_{D_i}$ denote the corresponding multiplication operator on $\mathcal{F}$, and let $m>0$ denote the degree sector under consideration. Generation conjecture. The joint eigenspaces for the operators $M_{D_i}$, $0\leq i\leq n$, are $1$-dimensional for all $m>0$. This conjecture would allow the divisor multiplication operators to determine multiplication by arbitrary classes, thereby extending the known equivalences of three-point functions with divisor insertions to general $r$-point functions via the degeneration formula.