Hachimori–Lim rank-growth conjecture for elliptic curves over imaginary quadratic fields

Let $E$ be an elliptic curve over an imaginary quadratic field $F$ satisfying the conditions stated in the source, and let $F^{(n)}$ and $F_{\operatorname{cyc}}$ denote the corresponding layers and cyclotomic extension. Hachimori–Lim's rank-growth conjecture. The conjecture asserts that

$$\operatorname{rank} E(F^{(n)})\leq \operatorname{rank} E(F_{\operatorname{cyc}})p^n.$$

This is identified in the source as a special case of a conjecture of Hachimori and Lim. The supplied passage does not state whether the conjecture has been resolved.