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.
The OP does not ask for advise, but quite generally, I would not recommend choosing for a Ph.D. project a problem that for a decade has resisted solution by experts.