The analogue of Coates–Sujatha's fine Selmer conjecture over function fields

Let $K$ be the function field considered in the paper, let $L$ be a finite extension of $K$, and let $L_\mathrm{cyc}$ be its cyclotomic extension. For an elliptic curve $E/L$, let $R(E/L_\mathrm{cyc})$ denote the fine Selmer group and let $R(E/L_\mathrm{cyc})^{\vee}$ be its Pontryagin dual. The analogue of Conjecture A. For every elliptic curve $E/L$,

$$R(E/L_\mathrm{cyc})^{\vee}\text{ is a finitely generated }\mathbb Z_p\text{-module}.$$

This is proposed as the function-field analogue of the number-field fine Selmer conjecture. The surrounding text places it in the characteristic-$p$ function-field setting and reports results establishing analogous statements in the cases treated by the paper.