Ext-analogues of Gan--Gross--Prasad models vanish for tempered representations

In this paper we prove that Ext-analogues of Gan–Gross–Prasad models vanish for tempered representations, as conjectured by D. Prasad. In particular, this implies a conjectural Euler–Poincaré characteristic formula for Gan–Gross–Prasad models. Besides, we also single out a special class of tempered representations, which includes the Steinberg representation, and show that the restrictions of them are projective.