Property (TTT) and the Cowling--Haagerup constant

We show that Property (TTT) is an obstruction to weak amenability with Cowling--Haagerup constant $1$. More precisely, if G is a countable group and H is an infinite subgroup of G such that the pair $(G,H)$ has relative Property (TTT), then the weak Haagerup constant ΛWH(G) is strictly greater than $1$. We apply this result to some semidirect products and lattices in higher rank algebraic groups.