Status of the Kashuba–Mathieu conjectures for free Jordan superalgebras with at most one even generator

Determine whether the Kashuba–Mathieu homology vanishing and character formula statements hold for free Jordan superalgebras Jord(x_1,\ldots,x_m \mid y_1,\ldots,y_n) when m ≤ 1; specifically, ascertain whether for the associated Tits–Allison–Gao Lie superalgebra TAG(Jord(x_1,\ldots,x_m \mid y_1,\ldots,y_n)) the sl₂-module H_k(TAG(\Jord(\cdot)), k) contains no trivial or adjoint component for all k>1, and whether the corresponding Grothendieck ring character identities for Jord and B(Jord) predicted by the Kashuba–Mathieu framework hold in this superalgebra setting.

Background

The authors disprove the Kashuba–Mathieu conjecture for free Jordan algebras and show failure for free Jordan superalgebras whenever the number of even generators m>1. Via an operadic argument, validity for algebras would imply validity for superalgebras, but since the conjecture fails in general, only restricted cases might survive.

They explicitly point out that their results do not resolve the case of superalgebras with at most one even generator, leaving open whether the vanishing of trivial/adjoint components in Lie (super)algebra homology and the associated character formulas remain valid in this regime.