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We prove a conjecture of Dipendra Prasad on the Ext-branching problem from$\mathrm{GL}_{n+1}(F)$ to $\mathrm{GL}_n(F)$, where $F$ is a $p$-adic field.

A vanishing Ext-branching theorem for $(\mathrm{GL}_{n+1}(F), \mathrm{GL}_n(F))$