Let $\Gamma$ be a Gromov hyperbolic group coming endowed with a word metric coming from some finite generating set. Let $\nu$ be an associated Patterson-Sullivan measure (quasi-conformal density).
I am interested, for each $g\in \Gamma$, the continuity properties of the Radon-Nikodym derivative $\frac{d\nu\circ g}{d\nu}$ as a function on the Gromov boundary $\partial \Gamma$ (the set of equivalence classes of geodesic rays, where two are equivalent of they stay a bounded distance apart).
Certainly, if $\Gamma$ is free, and a free generating set is chosen, then the word metric is $CAT(-1)$ and so the Busemann functions are continuous which implies that $\nu$ is actually conformal and continuous.
What about when $\Gamma$ is not free? The Busemann functions for a word metric on $\Gamma$ are then discontinuous.
Does it necessarily follow that for each $g\in \Gamma$, the function $\zeta\to \frac{d\nu\circ g}{d\nu}(\zeta)$ must be discontinuous on $\partial \Gamma$?.
Moreover, in Remark 2.15 of the paper "ENTROPY AND DRIFT IN WORD HYPERBOLIC GROUPS" by Gouezel, Matheus, and Maucourant the authors claim that "for nice groups, such as surface groups this function $\frac{d\nu\circ g}{d\nu}$ is continuous at all but finitely many points" of $\partial \Gamma$.
http://web.univ-ubs.fr/lmam/matheus/articles/hlv.pdf
Can anyone provide a proof or reference for this fact, or some clarification of what "nice" groups are?