One-dimensionality conjecture for the cohomology of conformally invariant loop cocycles

A configuration is either a pair $(\ell,\Sigma)$ consisting of a Riemann surface $\Sigma$ and a loop $\ell\subset\Sigma$, or a triple $(\ell,\Sigma_1,\Sigma_2)$ consisting of Riemann surfaces $\Sigma_1\subset\Sigma_2$ and a loop $\ell\subset\Sigma_1$. Let $\mathcal{C}$ be the set of real-valued, conformally invariant, loop-continuous additive cocycles $f(\ell,\Sigma_1,\Sigma_2)$, and let $\mathcal{B}$ be the set of coboundaries of the form

$$f(\ell,\Sigma_1,\Sigma_2)=g(\ell,\Sigma_2)-g(\ell,\Sigma_1),$$

where $g$ is conformally invariant and continuous in $\ell$. Define $\mathcal{H}=\mathcal{C}/\mathcal{B}$. The cohomology one-dimensionality conjecture. The cohomology group $\mathcal{H}$ is a one-dimensional real vector space. This would classify the absolute-continuity classes of the MKS family of measures by showing that restriction-function cocycles have, modulo coboundaries, a single independent class. The supplied text does not state whether this claim has been proved or disproved.