Vanishing conjecture for perturbed Dirac cohomology

Let ${\mathfrak g}$ be a Lie superalgebra, ${\mathfrak l}$ a subalgebra, $M$ a ${\mathfrak g}$-supermodule, and $x\in {\mathcal Y}_{{\mathfrak l}}$. Write ${\operatorname{H}}_{\operatorname{D}_{{\mathfrak{g}},{\mathfrak l}}}(M)$ for the Dirac cohomology of $M$, ${\operatorname{DS}}_{x}$ for Duflo--Serganova cohomology, and ${\operatorname{H}}_{\operatorname{D}^{x}_{{\mathfrak{g}},{\mathfrak l}}}(M)$ for the cohomology of the perturbed Dirac operator. Vanishing conjecture. If

$$\operatorname{DS}_{x}\bigl({\operatorname{H}}_{\operatorname{D}_{{\mathfrak{g}},{\mathfrak l}}}(M)\bigr)=0,$$

then

$${\operatorname{H}}_{\operatorname{D}^{x}_{{\mathfrak{g}},{\mathfrak l}}}(M)=0.$$

For unitarizable modules this vanishing follows from the established identification of perturbed Dirac cohomology with Duflo--Serganova cohomology of ordinary Dirac cohomology; the conjecture concerns the non-unitarizable case and is suggested by a spectral-sequence argument.