Independence conjecture for the vanishing order of P-adic L-series

Let $E$ be the Anderson $t$-module under consideration, let $P$ vary over primes of $A$, and define the order of vanishing of the $P$-adic $L$-series at $z=1$ by

$$\operatorname{ord}_{z=1}L_P(\widetilde{E}/\widetilde{\mathscr{O}_L})$$

to be the greatest integer $n$ such that $(z-1)^n$ divides $L_P(\widetilde{E}/\widetilde{\mathscr{O}_L})$. Independence conjecture. The vanishing order of the $P$-adic $L$-series at $z=1$ is independent of $P$.

Caruso and Gazda had already conjectured this in the context of Anderson motives. The claim was proved in the case $L=K$ and $d=1$, while the general case remains open.