Orthogonality conjecture for points on the cubic elliptic-curve family

Let $a\geq 1$ be an integer and $0\neq d\in\mathbb Q$ such that $f(X)+d^2$ has distinct roots, where $f$ and its roots $\rho,\rho_2,\rho_3$, the splitting field $L_f$, and the elliptic curve $E$ are as in Theorem~. Define $P_1=(\rho,d,1)$, $P_2=(\rho_2,d,1)$, and $P_3=(\rho_3,d,1)$. The orthogonality conjecture. For $1\leq i\neq j\leq 3$, the points $P_i+P_j$ and $P_i-P_j$ are always orthogonal for the canonical height pairing on $E(L_f)$; in particular, $E(L_f)$ has rank at least two. The claim is motivated by computations and follows a theorem giving rank at least two under different hypotheses; the source does not establish the asserted universal orthogonality.