The injectivity conjecture for Anderson t-module exponentials

Let $E$ be the Anderson $t$-module under consideration over the extension $L/K$, let $P$ be a prime of $A$, and let $L_P(E/\mathscr{O}_L)$ denote its $P$-adic $L$-series. The exponential map is

$$\exp_E:L_\infty^d\rightarrow L_\infty^d.$$

Injectivity conjecture. The $P$-adic $L$-series is non-zero if and only if the exponential map $\exp_E:L_\infty^d\rightarrow L_\infty^d$ is injective.

The preceding proposition and theorem establish the implication from non-injectivity to vanishing. The converse is stated as a belief and is known when $d=1$ and $L=K$; it would characterize the totally real case through non-vanishing of the $P$-adic $L$-series.