Cartan subalgebras of tensor products of free quantum group factors with arbitrary factors

Let $\mathbb{G}$ be a free (unitary or orthogonal) quantum group. We prove that for any non-amenable subfactor $N\subset L^{\mathrm{\infty }}(\mathbb{G})$, which is an image of a faithful normal conditional expectation, and for any $\sigma$-finite factor $B$, the tensor product $N\otimes B$ has no Cartan subalgebras. This generalizes our previous work that provides the same result when $B$ is finite. In the proof, we establish Ozawa–Popa and Popa–Vaes’s weakly compact action on the continuous core of $N\otimes B$ as the one relative to B, by using an operator valued weight to B and the central weak amenability of $\mathbb{G}$.