Kolyvagin's nontriviality conjecture for modular forms

Let $N\geq 3$ and $k\geq 2$ be even integers, let $K$ be an imaginary quadratic field, let $f\in S_k(\Gamma_0(N))$ be a newform of trivial character, and let $p\nmid 6N$ be a rational prime that splits in $K$ at which $f$ is non-ordinary. With the associated representations, lattices, Selmer groups, Kolyvagin primes, Kolyvagin integers, and classes $\kappa_f(M,c)\in H^1(K,A_{f,M})$ as defined in the paper, write $\kappa_f$ for the set of all these classes.

Kolyvagin's conjecture. The set of Kolyvagin classes is nontrivial:

$$\kappa_f\neq\{0\}.$$

This is the modular-form version of Kolyvagin's conjecture for elliptic curves. The ordinary-prime case was proved by Zhang, while the paper establishes a version at non-ordinary primes under additional Selmer-rank hypotheses and then derives the full version.