Positive-capacity Julia set conjecture for quasiregular maps

Suppose that $d \geq 2$ and that $f: \mathbb{R}^d \to \mathbb{R}^d$ is a quasiregular map of transcendental type, meaning that $f$ has an essential singularity at infinity. Let $J(f)$ be the Julia set of $f$, and write $\operatorname{cap} S = 0$ when $S$ has zero conformal capacity and $\operatorname{cap} S > 0$ otherwise.

Positive-capacity Julia set conjecture. For every such map,

$$\operatorname{cap} J(f) > 0.$$

The conjecture arises because no examples are known in which the Julia set of a quasiregular map of transcendental type fails to have positive capacity. The Julia set is known to be infinite, closed, and completely invariant, but its positive capacity is not established in general.