The strict-decrease conjecture for asymptotic conformal graph energies

Let $(\pi,\phi)$ be a recurrent virtual graph endomorphism, and for $q\in[1,\infty]$ let $\overline{E}{}^q[\pi,\phi]$ denote its asymptotic $q$-conformal graph energy. Strict-decrease conjecture. The function

$$q\longmapsto \overline{E}{}^q[\pi,\phi]$$

is either constant or strictly decreasing. The preceding discussion establishes continuity and monotonicity of this energy function and places the Ahlfors-regular conformal dimension between its critical exponents; the proposed assertion would further determine the form of its level set, but no resolution is supplied here.