Property $(\mathrm{T}\mathrm{T}\mathrm{T})$ and the Cowling--Haagerup constant

We show that Property $(\mathrm{T}\mathrm{T}\mathrm{T})$ 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 $(\mathrm{T}\mathrm{T}\mathrm{T})$, then the weak Haagerup constant ${\mathbf{\Lambda }}_{\mathrm{W}\mathrm{H}}(G)$ is strictly greater than $1$. We apply this result to some semidirect products and lattices in higher rank algebraic groups.