The conformal-dimension characterization of Sierpiński carpet energy exponents

Let $(K,d)$ be the Sierpiński carpet under consideration, let $\dim_{\textup{ARC}}(K,d)$ denote its Ahlfors regular conformal dimension, and let $\mathcal{F}_{p}$ be the domain associated with the constructed $p$-energy. Then

Conformal-dimension characterization conjecture.

$$\dim_{\textup{ARC}}(K, d) = \inf\{ p \mid \mathcal{F}_{p}\text{ is embedded in a subset of }\mathcal{C}(K).\}$$

This conjecture proposes that the Ahlfors regular conformal dimension is exactly the threshold detected by embedding the $p$-energy domain into the space of continuous functions on the carpet. The paper identifies the inequality $p>\dim_{\textup{ARC}}(K,d)$ with the relevant Hölder-embedding regime, while the asserted characterization remains a big open problem for future work.