Restriction for general linear groups: the local non-tempered Gan-Gross-Prasad conjecture (non-Archimedean case)

We prove a local Gan-Gross-Prasad conjecture on predicting the branching law for the non-tempered representations of general linear groups in the case of non-Archimedean field. We also generalize to Bessel and Fourier-Jacobi models and study a possible generalization to Ext-branching laws.