Conjecture on conformal dimensions under decomposition by multicurves

Suppose $f\colon (S^2,A)\righttoleftarrow$ is a post-critically finite hyperbolic-type branched covering without a Levy cycle. Let $\Gamma$ be a completely $f$-invariant multicurve, and let $f_i\colon (S^2(i),A(i))\righttoleftarrow$ for $i=1,\dots,n$ be the small branched coverings of $(f,\Gamma)$, with first return times $\tau_i$. Write $\mathcal{J}_f$ and $\mathcal{J}_{f_i}$ for the corresponding Julia sets. Decomposition conformal-dimension conjecture. In the setting above, the Ahlfors-regular conformal dimensions satisfy

$$\operatorname{ARC.dim}(\mathcal{J}_f)\geq\max\left\{\operatorname{ARC.dim}(\mathcal{J}_{f_i})\mid 1\leq i\leq n\right\}.$$

This is presented as a consequence of the strict-decrease conjecture together with the monotonicity theorem for conformal energies under multicurve decompositions; no proof or resolution is supplied here.