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

We show that if (X^, ^) is a Bernoulli action of an i. c. c. nonamenable group which is weakly amenable with Cowling-Haagerup constant 1, and (Y, ) is a free ergodic p. m. p. algebraic action of a group, then the isomorphism L^ (X^) L^ (Y) implies that L^ (X^) 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.