Non-tempered Ext Branching Laws for the $p$-adic General Linear Group

Let $F$ be a non-archimedean local field. Let ${\pi }_1$ and ${\pi }_2$ be irreducible Arthur type representations of ${\mathrm{G}\mathrm{L}}_n(F)$ and ${\mathrm{G}\mathrm{L}}_{n-1}(F)$ respectively. We study Ext branching laws when ${\pi }_1$ and ${\pi }_2$ are products of discrete series representations and their Aubert-Zelevinsky duals. We obtain an Ext analogue of the local non-tempered Gan-Gross-Prasad conjecture in this case.