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

We show that if $\Gamma\curvearrowright (X^\Gamma,\mu^\Gamma)$ is a Bernoulli action of an i.c.c. nonamenable group $\Gamma$ which is weakly amenable with Cowling-Haagerup constant $1$, and $\Lambda\curvearrowright(Y,\nu)$ is a free ergodic p.m.p. algebraic action of a group $\Lambda$, then the isomorphism $L^\infty(X^\Gamma)\rtimes\Gamma\cong L^\infty(Y)\rtimes\Lambda$ implies that $L^\infty(X^\Gamma)$ and $L^\infty(Y)$ 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 \cite{BIP21}.