  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.
