On the Jacquet Conjecture on the Local Converse Problem for p-adic GL_n

Based on previous results of Jiang, Nien and the third author, we prove that any two minimax unitarizable supercuspidals of GL_N that have the same depth and central character admit a special pair of Whittaker functions. This result gives a new reduction towards a final proof of Jacquet's conjecture on the local converse problem for GL_N. As a corollary of our result, we prove Jacquet's conjecture for GL_N, when N is prime.