Local Heegner-point infinitude conjecture

Let $K$ be an imaginary quadratic field, let $p\geq 5$ be a prime, let $E$ be an elliptic curve over $\mathbb{Q}$ with modular parametrization $\pi$, and let $K_{\infty}/K$ be the anticyclotomic $\mathbb{Z}_p$-extension. For a prime $\mathfrak{p}_{\infty}$ of $K_{\infty}$ above $p$, write $K_{\mathfrak{p}_{\infty}}$ for the union of the corresponding completions, and let $\alpha_n$ be the Heegner points. The local Heegner-point infinitude conjecture. Suppose that $(E,\pi,p)$ satisfies $(*)$. If $E$ has ordinary reduction at $p$, there exists a prime $\mathfrak{p}_{\infty}$ above $p$ such that the $\Gamma$-submodule of $E(K_{\mathfrak{p}_{\infty}})/p$ generated by the $\alpha_n$ has infinite cardinality. If $E$ has supersingular reduction at $p$ and $p$ splits in $K/\mathbb{Q}$, there exists a prime $\mathfrak{p}_{\infty}$ above $p$ such that each of the $\Gamma$-submodules generated by the even-indexed points $\alpha_{2n}$ and by the odd-indexed points $\alpha_{2n+1}$ has infinite cardinality. The source presents this as a third conjecture and later uses its supersingular part to derive consequences for Selmer groups; no proof of the full statement is given.