Let $p$ be an odd prime and $K$ an imaginary quadratic field where $p$ splits. Under appropriate hypotheses, Bertolini showed that the Selmer group of a $p$-ordinary elliptic curve over the anticyclotomic $\mathbb Z_p$-extension of $K$ does not admit any proper $\Lambda$-submodule of finite index, where $\Lambda$ is a suitable Iwasawa algebra... We generalize this result to the plus and minus Selmer groups (in the sense of Kobayashi) of $p$-supersingular elliptic curves. In particular, in our setting the plus/minus Selmer groups have $\Lambda$-corank one, so they are not $\Lambda$-cotorsion. As an application of our main theorem, we prove results in the vein of Greenberg-Vatsal on Iwasawa invariants of $p$-congruent elliptic curves, extending to the supersingular case results for $p$-ordinary elliptic curves due to Hatley-Lei. (read more)