  Let $E/\mathbb{Q}$ be an elliptic curve and $p$ an odd prime. In 1991
Kolyvagin conjectured that the system of cohomology classes for the $p$-adic
Tate module of $E$ derived from Heegner points on $E$ over ring class fields of
an imaginary quadratic field $K$ is non-trivial. In this paper we prove
Kolyvagin's conjecture in the cases where $p$ is a prime of good ordinary
reduction for $E$ that splits in $K$ and a $p$-adic anticyclotomic Iwasawa Main
Conjecture for $E/K$ holds. In particular, our results cover many cases where
$p$ is an Eisenstein prime, complementing W. Zhang's progress towards the
conjecture.
  Our methods also yield a proof of a refinement of Kolyvagin's conjecture
expressing the divisibility index of the $p$-adic Heegner point Kolyvagin
system in terms of the Tamagawa numbers of $E$, as conjectured by W. Zhang in
2014, as well as proofs of analogous results for the Kolyvagin systems obtained
from the cyclotomic Euler system of Beilinson--Kato elements.
