Validity of the dichotomy for lower conformal dimension

Determine whether the dichotomy that the conformal dimension of every nonzero locally finite Borel measure is either zero or infinite remains valid when conformal dimension is defined using lower Hausdorff dimension rather than full-measure (upper) Hausdorff dimension.

Background

The paper defines the Hausdorff dimension of a measure using the infimum of the Hausdorff dimensions of Borel sets of full measure; this is commonly called the upper Hausdorff dimension of a measure. Using quasisymmetric push-forwards, the authors prove that the resulting conformal dimension of every nonzero locally finite Borel measure is either zero or infinite.

The authors then note an alternative definition based on lower Hausdorff dimension, obtained by taking the infimum of the Hausdorff dimensions of Borel sets having positive measure. They state that the corresponding analogue of the main zero-or-infinity theorem is unknown, although the analogues of the doubling-space corollary and the dimension-reduction theorem do hold under the lower-dimensional definition.