Remove the lower density assumption from the main theorem

Ascertain whether Theorem 1.1 (Theorem \ref{th:main}) remains valid without assuming positive lower p-density almost everywhere; specifically, determine if for any compact metric space (X,d) with 0 < H^p(X) < ∞ that is not p-rectifiable, there still exists a positive-measure subset F ⊂ X, a metric space (Y,d), and a Lipschitz map f: F → Y such that H^p_d(f(F)) > 0 and f has no biLipschitz restriction on any positive-measure subset of F, even when the hypothesis θ_*^p(X,x) > 0 almost everywhere is removed.

Background

The main theorem (Theorem \ref{th:main}) requires that the space has positive lower p-density almost everywhere to construct a Lipschitz map with positive-measure image and no positive-measure biLipschitz pieces. The authors indicate the lower density is used at a specific point (via equation (\ref{eq:Fr-defn}) in proving Lemma \ref{l:not-bilip}).

They ask whether the theorem’s conclusion still holds without this density hypothesis, which would broaden the applicability of their characterization.