Uniform measure equivalence invariance conjecture for $L^2$-torsion

Let $G$ and $H$ be countable groups such that all the $L^2$-Betti numbers of $G$ and $H$ vanish. Assume that both $G$ and $H$ admit finite CW-models for their classifying spaces. They satisfy the determinant conjecture. Two groups are measure equivalent with index $c=I(G,H)>0$ if they admit a measure coupling of that index. Uniform measure equivalence invariance conjecture. If $G$ and $H$ are measure equivalent with index $c$, then

$$\rho^{(2)}(G)=c\cdot\rho^{(2)}(H).$$

This conjecture predicts that $L^2$-torsion transforms proportionally under measure equivalence, analogously to the invariance of $L^2$-Betti numbers. The paper presents it as conjectural; the supplied text gives no resolution.