The Ext-branching conjecture for Arthur type representations of general linear groups

Let $G_n=\mathrm{GL}_n(F)$, and let $\pi_M$ and $\pi_N$ be Arthur type representations of $G_{n+1}$ and $G_n$, respectively. For $k\geq 1$, write $\pi_M^{[k]}$ and ${}^{(k-1)}\pi_N$ for the derived representation-theoretic operations used in the source. Ext-branching conjecture. For any $i$,

$$\mathrm{Ext}^i_{G_n}(\pi_M,\pi_N)\cong\bigoplus_k\mathrm{Ext}^i_{G_{n+1-k}}(\pi_M^{[k]},{}^{(k-1)}\pi_N).$$

This is proposed as a possible generalization of observations on branching laws; the supplied text does not state a proof or resolution.