The positive-capacity conjecture for completely invariant sets

Let $f:\overline{{\mathbb R}^n}\to\overline{{\mathbb R}^n}$ be quasiregular with $\operatorname{deg}(f)>K_I(f)$. Let $E(f)$ be the exceptional set. A set $X$ is completely invariant when $f(X)=X=f^{-1}(X)$. Let $\operatorname{cap}X$ denote its capacity. Positive-capacity conjecture. If

$$X\subset\overline{{\mathbb R}^n}\setminus E(f)$$

is compact and completely invariant, then

$$\operatorname{cap}X>0.$$

The paper explains that this conjecture would imply the Lipschitz-free versions of the orbit and Julia-set theorems. It is presented as an open conjecture about invariant sets in quasiregular dynamics.