Cowling's conjecture on weak amenability constant one and the Haagerup property

Let $G$ be a locally compact group. The one-way Cowling conjecture. If $G$ is weakly amenable with $\boldsymbol\Lambda(G)=1$, then it has the Haagerup property.

This is the direction of the original Cowling conjecture that remains open after the converse implication was disproved. It asks whether weak amenability with Cowling--Haagerup constant one forces the existence of a net of positive definite functions in $C_0(G)$ converging uniformly to $1$ on compact subsets.