Equality of associated varieties for irreducible highest-weight modules

Let $\frak g$ be a contragredient simple Lie superalgebra. For an integral dominant weight $\lambda\in\frak h^*$, let $L_\lambda$ be the corresponding irreducible highest-weight module, let $k$ be the degree of atypicality of $\lambda$, and let $\bar{X}_k$ denote the closure of the rank-$k$ associated-variety stratum. Write $X_{L_\lambda}$ for the associated variety of $L_\lambda$.

Associated-variety conjecture. For any integral dominant $\lambda\in\frak h^*$ with degree of atypicality $k$,

$$X_{L_\lambda}=\bar{X}_k.$$

This predicts that the associated variety of every irreducible highest-weight module is determined precisely by the degree of atypicality. The supplied text gives the assertion as a conjecture but provides no resolution status, so it remains open here.