For F4, the BSDH variety from a Coxeter-element power is rigid exactly when a1 is not 3 or a2 is not 2; for G2 it is never rigid.
Demazure, A very simple proof of Bott’s theorem, Invent
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Rigidity of Bott-Samelson-Demazure-Hansen variety for $F_4$ and $G_2$
For F4, the BSDH variety from a Coxeter-element power is rigid exactly when a1 is not 3 or a2 is not 2; for G2 it is never rigid.