  Let $p\ge 5$ be a prime number, $E/\mathbb{Q}$ an elliptic curve with good
supersingular reduction at $p$ and $K$ an imaginary quadratic field such that
the root number of $E$ over $K$ is $+1$. When $p$ is split in $K$, Darmon and
Iovita formulated the plus and minus Iwasawa main conjectures for $E$ over the
anticyclotomic $\mathbb{Z}_p$-extension of $K$, and proved one-sided inclusion:
an upper bound for plus and minus Selmer groups in terms of the associated
$p$-adic $L$-functions. We generalize their results to two new settings:
  1. Under the assumption that $p$ is split in $K$ but without assuming
$a_p(E)=0$, we study Sprung-type Iwasawa main conjectures for abelian varieties
of $\mathrm{GL}_2$-type, and prove an analogous inclusion.
  2. We formulate, relying on the recent work of the first named author with
Kobayashi and Ota, plus and minus Iwasawa main conjectures for elliptic curves
when $p$ is inert in $K$, and prove an analogous inclusion.
