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

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