Commensurability invariance of property (LR) for hyperbolic groups

Let $G$ be a hyperbolic group, and suppose that $G$ has property (LR). Let $H$ be a finite index supergroup of $G$. The commensurability conjecture for hyperbolic groups. The group $H$ has property (LR). A positive answer would imply property (LR) for all one-relator groups with torsion, since every such group has a finite index subgroup with (LR).