Strong Kolyvagin higher-weight non-vanishing conjecture

Let $\kappa^{\mathrm{st}}_{f,\infty}$ be the strict Kolyvagin set consisting of the classes $c_1(f,n)$ as $n$ ranges over the Kolyvagin indices. Strong Kolyvagin conjecture.

$$\kappa^{\mathrm{st}}_{f,\infty}\neq\{0\}.$$

This is the strong form obtained by requiring non-vanishing already at level $M=1$. The source introduces it as a stronger variant of the higher-weight Kolyvagin conjecture and gives no resolution.