The optimal removability conjecture for bounded quasiregular mappings

A closed set $E\subset \mathbb{R}^n$ is removable under bounded $K$-quasiregular mappings if every bounded $K$-quasiregular mapping defined on an open set minus $E$ extends to a $K$-quasiregular mapping on the whole open set. Let $d=n/(K+1)\leq n/2$. Optimal removability conjecture. Sets of Hausdorff $d$-measure zero are removable under bounded $K$-quasiregular mappings. The preceding theorem establishes removability for sets of some positive Hausdorff dimension depending on $K$; this conjecture proposes the optimal threshold suggested by conjectures concerning $p$-norms of the Hilbert transform on forms and the relationship between $s$-capacity and Hausdorff dimension.