Density version of Lerman's $L^2$ curve learning theorem

Let $\mu$ be a locally finite Borel measure on $\mathbb{R}^n$, let $Q_1\in\vec{\Delta}(\mathbb{R}^n)$, and let $\lambda>1$ be the constant from Lerman's theorem. Define

$$A=\{x\in\lambda Q_1:\widehat{J}_2(\mu,x)\leq\varepsilon\}.$$

Suppose that $\varepsilon\leq\varepsilon_0(n)$ and $\delta\geq\delta_0(n,\varepsilon)$ satisfy $\mu(A)\geq\delta\mu(\lambda Q_1)$. Density version of Lerman's theorem. A version of Lerman's theorem should hold under this hypothesis, with a rectifiable curve $\Gamma_1\subset\lambda Q_1$ satisfying

$$\mu(A\cap\Gamma_1)\geq C^{-1}e^{-C\varepsilon}\mu(Q_1),$$

where $C=C(n,\delta)$. This would provide the density analogue needed to extend the curve-learning strategy from uniform control of the Jones function to almost-everywhere finiteness; the source does not indicate that this version is known.