A unique Cartan subalgebra result for Bernoulli actions of weakly amenable groups

We show that if Γ (X^Γ,μ^Γ) is a Bernoulli action of an i.c.c. nonamenable group Γ which is weakly amenable with Cowling-Haagerup constant $1$, and Λ(Y,ν) is a free ergodic p.m.p. algebraic action of a group Λ, then the isomorphism L^∞(X^Γ)Γ L^∞(Y)Λ implies that L^∞(X^Γ) and L^∞(Y) are unitarily conjugate. This is obtained by showing a new rigidity result of non properly proximal groups and combining it with a rigidity result of properly proximal groups from {BIP21}.