The mock plectic invariant rank-distribution conjecture

Let $K$ be a quadratic imaginary field, let $E/\mathbb Q$ be the elliptic curve in the construction, let $E'$ be its quadratic twist attached to $K$, and let $\text{Capricorn}(\tau)$ denote the mock plectic invariant. Mock plectic invariant rank-distribution conjecture. If $L(E/K,1)=0$, then

$$\text{Capricorn}(\tau)\ne0\quad\Longleftrightarrow\quad \begin{cases} r_{\mathrm{alg}}(E/\mathbb Q)=0,\ r_{\mathrm{alg}}(E'/\mathbb Q)=2,&\text{if }a_p(E)=+1,\\ r_{\mathrm{alg}}(E/\mathbb Q)=1,\ r_{\mathrm{alg}}(E'/\mathbb Q)=1,&\text{if }a_p(E)=-1. \end{cases}$$

This conjecture predicts when the mock plectic invariant is nonzero in the rank-two setting, with the distribution between the two quadratic eigenspaces governed by the reduction type at $p$. Its status is not resolved in the source.