Bonk–Kleiner's conformal-dimension characterization of Cannon's conjecture

Let $\Gamma$ be a hyperbolic group whose boundary is homeomorphic to the $2$-sphere. The conformal dimension of $\partial\Gamma$ is the infimum of the Hausdorff dimensions of metrics in its conformal gauge.

Bonk–Kleiner's conjecture. The conformal dimension of $\partial\Gamma$ is attained by a metric in its conformal gauge.

Bonk and Kleiner showed that this assertion is equivalent to Cannon's conjecture. The statement is therefore open in general, although it is known in the cases where Cannon's conjecture is known.