Surjectivity conjecture for the graded evaluation map of hyperelliptic Abelian functions

Let $A$ be the algebra of hyperelliptic Abelian functions, let ${\cal D}$ be the differential-operator algebra acting on $A$, and let $W^g$ be the relevant exterior-power space. With the grading and homogeneous components defined by

$$\text{deg}\,du_i=-(2i-1),$$

write

$$\text{ev}^{gr}:{\cal D}\otimes W^g\longrightarrow \text{gr}^{KP}\,A\otimes\wedge^g T^*$$

for the graded evaluation map. For $I=(i_1,\ldots,i_r)$ and $J=(i_{r+1},\ldots,i_g)$, let $J^c$ be the complementary index set and $(I;J^c)$ the corresponding determinant of the $\eta]_{ij}$.

Surjectivity conjecture. The map $\text{ev}^{gr}$ is surjective. Equivalently, $\text{gr}^{KP}\,A$ is generated as a ${\cal D}$-module by $1\in \text{gr}^{KP}_0A$ and the elements $(I;J^c)\in \text{gr}^{KP}_{d_{I,J}+g^2}A$ for $r\geq1$, with $I=(i_1,\ldots,i_r)\in\{1,\ldots,g\}^r$ and $J=(i_{r+1},\ldots,i_g)\in\{1,\ldots,g\}^{g-r}$.

The assertion gives a generation description for the Kazhdan–Polishchuk graded algebra of hyperelliptic Abelian functions in terms of the differential operators and determinant generators. The supplied source attributes it to NS1, but provides no resolution status, so it is recorded as open.