Mestre bound sharpness conjecture over number fields

Fix a number field $F$. For each positive integer $N$, let $r_F(N)$ be the maximum of $\operatorname{Rank}_{\mathbb{Z}}E(F)$ over elliptic curves $E/F$ whose conductor $\mathfrak{n}_E$ has norm $N$, and set $r_F(N)=0$ if no such curve exists. Mestre bound sharpness conjecture. One has

$$\limsup_N\frac{r_F(N)}{\log N/\log\log N}>0.$$

This conjecture asserts that the order of growth suggested by Mestre’s upper bound is attained infinitely often, but the source provides no resolution.