On the inverse absolute continuity of quasiconformal mappings on hypersurfaces

We construct quasiconformal mappings $f\colon \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ for which there is a Borel set $E \subset \mathbb{R}^2 \times \{0\}$ of positive Lebesgue $2$-measure whose image $f(E)$ has Hausdorff zero. This gives solution to the open problem inverse absolute continuity on hypersurfaces, attributed Gehring. By implication, our result also answers questions V\"ais\"al\"a and Astala--Bonk--Heinonen.