Bremner's conjecture on additive rigidity of elliptic-curve points

Let $E/\a0\mathbb{Q}$ be an elliptic curve of Mordell\a0\a0Weil rank $r$, and let $\{P_1,\ldots,P_N\}$ be a sequence of rational points whose $x$-coordinates form an arithmetic progression in $\mathbb{Q}$. Bremner's conjecture. There exists an absolute constant $A$ such that

$$N \ll A^r.$$

This precise formulation expresses the idea that long $x$-arithmetic progressions should force large Mordell\a0\a0Weil rank. It is supported by several results, and Garcia-Fritz and Pasten proved it for families with fixed $j$-invariant; the general conjecture remains open.