The positivity conjecture for 1-bounded entropy

Let $r>1$ and let $Q$ be a von Neumann subalgebra of the free group factor $L({\mathbb F}_r)$. Assume that $Q$ is diffuse and nonamenable. Positivity conjecture for 1-bounded entropy. The relative 1-bounded entropy satisfies

$$h(Q:L({\mathbb F}_r))>0.$$

This is stated as an intermediate conjecture that would imply the Peterson-Thom conjecture; the source does not provide evidence of a resolution.