Leopoldt conjecture for Anderson t-modules

Let $E$ be an Anderson $t$-module, let $P$ be a prime of $A$, and define

$$U(E;P\mathscr{O}_L)=\{x\in \operatorname{Lie}_E(L_\infty)\mid \exp_E(x)\in E(P\mathscr{O}_L)\}$$

and

$$\mathscr{U}(E;P\mathscr{O}_L)=\exp_E(U(E;P\mathscr{O}_L)).$$

Also write $\mathscr{U}(E;\mathscr{O}_L)=\exp_E(U(E;\mathscr{O}_L))$. Leopoldt conjecture. The $A_P$-rank of $\mathscr{U}(E;P\mathscr{O}_L)$ is equal to the $A$-rank of $\mathscr{U}(E;\mathscr{O}_L)$.

The claim is an analogue of Leopoldt's conjecture for the Carlitz module. It is clear when $d=1$ and $L=K$, and it was proved for the Carlitz module over the $P$th cyclotomic extension.