The finite basis problem for all subvarieties of the variety $\mathsf{V}(S_7^0)$

We solve the finite basis problem for all subvarieties of $\mathsf{V}(S_7^0)$. By considering their intersections with $\mathsf{V}(S_7)$, we reduce the problem to six intervals. We show that $\mathsf{V}(S_7^0)$ has exactly fifteen finitely based subvarieties and that every other subvariety is nonfinitely based. Moreover, $\mathsf{V}(S_c(abc))$ and $\mathsf{V}(S{R}_6)$ are the only limit subvarieties of $\mathsf{V}(S_7^0)$.