Honda's conjecture on linear rank growth of abelian varieties

Let $A$ be an abelian variety over a number field $k$.

Honda's conjecture. There is $c=c(A,k)>0$ such that for every finite extension $L/k$,

$$\operatorname{rank} A(L) \le c\cdot [L:k].$$

This conjecture predicts uniform linear growth of Mordell–Weil ranks under finite extensions and directly implies boundedness of ranks in quadratic twist families of a fixed elliptic curve. Its status is not specified in the supplied text.