Cohen–Macaulayness of Severi strata for miniversal deformations of $A_{2k}$

Let $A_{2k}$ be the indicated singularity, let $S$ be the base of its miniversal deformation, and let $D(\ell)\subset S$ denote the corresponding Severi stratum. For integers $\ell$ and $k$ with $\ell\leq k$, the Cohen–Macaulayness conjecture.

$$D(\ell)\text{ is Cohen--Macaulay.}$$

The statement is supported in the source by the displayed Betti-number data for $A_{2k}$ with $k\leq4$, which imply Cohen–Macaulayness in those cases. The assertion for all $\ell\leq k$ is left as a conjectural generalization.