de Weger's conjecture on large Tate–Shafarevich groups over function fields

Let $K=\mathbb{F}_q(t)$, and for an elliptic curve $E/K$ let $H(E)$ denote its exponential differential height, $N(E)$ its numerical conductor, and $\@math@cyr{\cyracc SH}(E)$ its Tate–Shafarevich group. Assume that the relevant Tate–Shafarevich groups are finite. de Weger's conjecture. For every $\varepsilon>0$, there are infinitely many elliptic curves $E/K$ such that

$$|\@math@cyr{\cyracc SH}(E)|\gg_{q,\varepsilon} H(E)^{1-\varepsilon},$$

and there are infinitely many elliptic curves $E/K$ such that

$$|\@math@cyr{\cyracc SH}(E)|\gg_{q,\varepsilon} N(E)^{1/2-\varepsilon}.$$

This is the function-field analogue of de Weger's conjectures over $\mathbb{Q}$, asserting that the exponents in the corresponding upper bounds are optimal up to $\varepsilon$.