  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 $K$, Castella and
Wan formulated the plus and minus Heegner point main conjecture for $E$ along
the anticyclotomic $\mathbb{Z}_p$-extension of $K$, and proved it for
semistable curves. We generalize their results to two settings:
  1. Under the assumption that $p$ is split in $K$ but without assuming
$a_p(E)=0$, we study Sprung-type main conjectures for $\mathrm{GL}_2$-type
abelian varieties at non-ordinary primes and prove it under some conditions.
  2. We formulate, relying on the recent work of the first-named author with
Kobayashi and Ota, plus and minus Heegner point main conjectures for elliptic
curves when $p$ is inert in $K$, and prove the minus main conjecture for
semistable elliptic curves.
  These results yield a $p$-converse to the Gross--Zagier and Kolyvagin theorem
for $E$. Our method relies on Howard's framework of bipartite Euler system,
Zhang's resolution of Kolyvagin's conjecture and the recent proof of
Kobayashi's main conjecture.
