TITLE: Is the p-adic Lindemann-Weierstrass Conjecture still open?
QUESTION [12 upvotes]: The p-adic Lindemann-Weierstrass Conjecture: Let $\alpha_{1},\ldots,\alpha_{N}\in\overline{\mathbb{Q}_{p}}$
  be distinct $p$-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.

REPLY [13 votes]: Here is a 2018 paper, A Note on One-dimensional Varieties Over the Complex p-adic Field, that still lists the "full" statement as a conjecture; "half" of the statement, meaning that at least $\lfloor N/2\rfloor$ of the exponents are independent, has been proven by Nesterenko.