The generalized Szpiro conjecture for abelian varieties

Let $A/K$ be an abelian variety of dimension $g$ over a number field $K$, with conductor ideal $\mathcal{F}_{A/K}$. Generalized Szpiro conjecture. There exist real numbers $c_1$ and $c_2$, depending at most on $g$ and $K$, such that

$$h_{Falt}(A/K)\leq c_1\log N_{K/\mathbb{Q}}(\mathcal{F}_{A/K})+c_2.$$

This is an inverse height–conductor inequality intended to turn the paper's Tate–Shafarevich bounds into bounds controlled by the conductor, rank, field, and dimension; the supplied text gives no resolution.