The one-sided fundamental semigroup conjecture for property (T) group factors

Let $G$ be an icc property (T) group. For a $\mathrm{II}_1$ factor $\mathcal M$, let $\mathcal F_s(\mathcal M)$ be the set of all $t\in\mathbb R_+$ for which there exists a $*$-embedding $\mathcal M\hookrightarrow\mathcal M^t$. One-sided fundamental semigroup conjecture. For every icc property (T) group $G$,

$$\mathcal F_s(\mathcal L(G))=\mathbb N.$$

This is presented as a strengthening of Popa's conjecture and concerns the possible one-sided fundamental semigroups of group von Neumann algebras; the supplied text leaves it open.