Planar self-similar set projections and fibres

Determine whether every line projection of a planar self-similar set with dense rotations and Hausdorff dimension $s>1$ has positive one-dimensional Lebesgue measure and fibres of Hausdorff dimension $s-1$ at Lebesgue-almost every point of the projected image.

Background

The main set theorem proves positive projection volume and almost-everywhere fibre dimension in dimensions $d\ge3$ under dense rotations. The planar analogue for the uniform measure is false because there are examples with singular projections and discrete conditional measures. However, singularity of a projected measure does not by itself imply that the support has zero Lebesgue measure, so the corresponding set statement remains unresolved. The group-mixing argument used in the paper cannot handle $SO(2)$ because it lacks the required compact semisimple structure.