Independence of the natural line from the quadratic field

Let $(E,p,K)$ be a triple in which $E$ is an elliptic curve over $\mathbf{Q}$ of analytic rank $2$, $p$ is an odd prime of anomalous good ordinary reduction for $E$, and $K$ is an imaginary quadratic field satisfying the Heegner hypothesis for $E/\mathbf{Q}$, with analytic rank of $E^K/\mathbf{Q}$ equal to $1$ and $p$ split in $K$. Suppose that $E(\mathbf{Q}_p)[p]$ is trivial, $E$ has no rational $p$-isogeny, $p$ does not divide the class number of $K$, and $R\in E(K)$ generates $E^K(\mathbf{Q})$ and is not $p$-divisible in $E(\mathbf{Q}_p)$. Let $H$ be the receptacle, let $L'_K$ be the shadow line, and let $\mathcal{L}$ be the natural line. Natural-line independence conjecture. The image of $L'_K$ in $H\otimes \mathbb{F}_p$ is independent of $K$ and coincides with $\mathcal{L}$. The additional local torsion, isogeny, class-number, and divisibility hypotheses are intended to isolate the stable filtered distribution observed in the numerical data; no resolution is supplied, so the conjecture is open.