Closure of approximate $L^2$-rigidity under joins with diffuse intersection

Let $M$ be a $\rm II_1$ factor and let $Q_1,Q_2\leq M$ be von Neumann subalgebras such that $Q_i\leq M$ is approximately $L^2$-rigid for $i=1,2$. Suppose that $Q_1\cap Q_2$ is diffuse. Approximate $L^2$-rigidity join conjecture. Then $Q_1\vee Q_2\leq M$ is approximately $L^2$-rigid.

This is the von Neumann algebra analogue of a result of Peterson and Thom for subgroups with infinite intersection. A positive solution would imply the stated amenability conjecture for subalgebras of $L(\mathbb F_2)$, but the source provides no resolution.