Sufficiency of the transformed Carleson condition for degenerate monotone transformations

Prove that if a strictly increasing continuous function \(F\) on \(J_0=[c_0^{-1},c_0]\) and an \(n\)-AD-regular measure \(\mu\) with \(\mu(\mathbb{R}^d)=\infty\) satisfy the Carleson condition for \(\Delta^F_\mu\), then \(\mu\) is uniformly \(n\)-rectifiable.

Background

The main theorem establishes equivalence between uniform rectifiability and the Carleson condition after composing density differences with a function that is bi-Lipschitz on the relevant density interval. For strictly increasing continuous functions that are not bi-Lipschitz, the direct pointwise comparison argument fails. The proposed route is to convert smallness of the transformed square function into small oscillation of the original density using uniform continuity of $F^{-1}$, then invoke the weak constant density criterion for uniform rectifiability. The authors state this result as a conjecture and note that the quantitative details have not been carried out.