Pilgrim–Thurston conjecture on strict decrease of conformal energy

Let $f\colon (S^2,A)\righttoleftarrow$ be a post-critically finite hyperbolic-type branched covering, and let $\overline{E}^p(f)$ denote its asymptotic $p$-conformal energy for $p\in[1,\infty]$. Pilgrim–Thurston's conjecture. The function $p\mapsto\overline{E}^p(f)$ is either constant or strictly decreasing on $[1,\infty]$. This conjecture would imply, together with the known identity $\overline{E}^{p_*}(f)=1$ for $p_* = \operatorname{ARC.dim}(\mathcal{J}_f)$, that the Ahlfors-regular conformal dimension of a Julia set can be determined by asymptotic conformal energies. Its resolution is not given here.