The horizontal nonvanishing conjecture for big Heegner points

Let $\mathfrak{X}_1$ be the big Heegner point of conductor $c=1$, and define

$$\mathfrak{Z}_0=\operatorname{Cor}_{H_1/K}(\mathfrak{X}_1)\in\tilde{H}^1_f(K,\mathbf{T}^\dagger).$$

Horizontal nonvanishing conjecture. The cohomology class $\mathfrak{Z}_0$ is not $R$-torsion.

This conjecture is the horizontal nonvanishing statement for the big Heegner point at conductor one. The paper explains that it would imply, via an extension of Kolyvagin's theory due to Nekovář, Greenberg's prediction that the Selmer-group dimension is zero or one for all but finitely many specializations in the Hida family.