Plectic points and the algebraic rank of quadratic twists

Let $A/F$ be a modular elliptic curve, let $E/F$ be a quadratic CM extension of class number one, let $A^E/F$ be the quadratic twist of $A$ by $E/F$, and let $\psi$ be an optimal embedding of conductor $1$. Write $r_{\mathrm{alg}}(A^E/F)$ for the algebraic rank. Plectic rank conjecture. There exists a non-torsion plectic point $\mathrm{Q}_{\psi,j}^{\mathbbm{1}}$ if and only if

$$r_{\mathrm{alg}}(A^E/F)=r.$$

Moreover, if the cited $\operatorname{Kol}$-type condition and the algebraicity conjecture hold, then the points $w_{\psi,1}^{\mathbbm{1}},\dots,w_{\psi,r}^{\mathbbm{1}}$ generate $A^E(F)$. This connects the non-torsion of the distinguished plectic points with the full algebraic rank of the quadratic twist and predicts generators of its rational points.