The p-adic Lindemann-Weierstrass Conjecture: Let $\alpha_{1},\ldots,\alpha_{N}\in\overline{\mathbb{Q}_{p}}$ be $p$-adic algebraic numbers satisfying $\left|\alpha_{n}\right|_{p}<p^{-\frac{1}{p-1}}$ (so that $\exp_{p}\left(\alpha_{n}\right)\in\mathbb{C}_{p}$) for all $n$. Then, $\exp_{p}\left(\alpha_{1}\right),\ldots,\exp_{p}\left(\alpha_{N}\right)$ are algebraically independent over $\mathbb{Q}$.

I'm a graduate student who is considering taking on this problem for my doctoral dissertation

This article from 2008 by M. Waldschmidt says that the conjecture is still open (it lists it as conjecture 5.16).

I was wondering if that was still the case.