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Structure of minimal singular vectors when U^pr is not simple

Determine the structure of the singular vector of minimal conformal weight in the universal principal W-algebra U^pr of V_k(psl(2|2)) at levels k where U^pr is not simple; specifically, prove that this singular vector has horizontal grade +1 and multiplicity 1.

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Background

The authors introduce a horizontal grading on Upr and numerically observe singular vectors at certain nonintegral rational levels. Beyond the simplicity conjecture, they further hypothesize a precise property of the lowest-weight singular vector at non-simple levels.

Confirming this property would refine knowledge of the ideal structure of Upr and guide the construction of its simple quotients and modules.

References

We also conjecture that when $\upr$ is not simple, then the singular vector of minimal conformal weight has grade $+1$ and multiplicity $1$.

— The principal W-algebra of $\mathfrak{psl}_{2|2}$ (2509.04795 - Fehily et al., 5 Sep 2025) in Section 2.3 (Operator product expansions), end of itemized observations