**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](https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/AWSLecture5.pdf) says that the conjecture is still open (it lists it as conjecture 5.16).


I was wondering if that was still the case.



Is the p-adic Lindemann-Weierstrass Conjecture still open?
Here is a 2018 paper, [A Note on One-dimensional Varieties Over the Complex p-adic Field](https://www.ejpam.com/index.php/ejpam/article/view/3281), 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.](http://math.tsukuba.ac.jp/ant/Sympo/dioph06_abst/Nesterenko.pdf)


