Conjecture on minors of the cubic discriminant matrix over singularity loci

Let $M$ be the matrix whose determinantal rank loci are denoted by $V_k$, and let $N_k$ denote the locus of cubic surfaces with $k$ singularities. For each $k$, consider the minors of size $20-k+1$. Conjecture on the minors of $M$. For $1\leq k\leq4$, the $(20-k+1)\times(20-k+1)$ minors of $M$ vanish on $N_k$. This pattern is suggested by the verified case of binodal cubic surfaces in normal form, but the assertion for the stated range is not established in the supplied text.