Here is a 2018 paper, <A HREF="https://www.ejpam.com/index.php/ejpam/article/view/3281">A Note on One-dimensional Varieties Over the Complex p-adic Field</A>, 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 <A HREF="http://math.tsukuba.ac.jp/ant/Sympo/dioph06_abst/Nesterenko.pdf">Nesterenko.</A>