Maximal submodule of V^1(psl_{n|n}) is generated by the singular vector U

Prove that the maximal proper submodule of the universal affine vertex algebra V^1(psl_{n|n}) at level 1 is the submodule U generated by the singular vector $\chi=E^{1,2n-1}_{(-1)}E^{1,2n}_{(-1)}\ket{0}$, and equivalently show that the simple affine quotient satisfies $L_1(psl_{n|n})=V^1(psl_{n|n})/U$.

Background

The authors construct a specific singular vector in V^1(psl_{n|n}) and the submodule U it generates, use it to constrain the associated variety, and ultimately prove the main geometric result of the paper. They conjecture that U is the maximal proper submodule, which would give a precise structural description of the simple quotient.

They outline potential approaches to proving this conjecture, including inverse quantum Hamiltonian reduction and extending related results in the literature.