Equality of Segre–Stiefel–Whitney classes for Morin and corank-two singularities

Let $l$ be an integer, and let $\overline{A_2}(l+1)$ and $\overline{\Sigma}^2(l)$ denote the corresponding singularity loci. Their Segre–Stiefel–Whitney classes are defined in the setting of the paper. Segre–Stiefel–Whitney class equality conjecture. The Segre-SW classes of $\overline{A_2}(l+1)$ and $\overline{\Sigma}^2(l)$ are equal. This conjecture is motivated by computations showing equality up to degree $8$ for $l=0$ and up to degree $12$ for $l=1$, but no general proof is given.