The disjoint-interval density conjecture

Let $\mu$ be an absolutely continuous probability measure supported on $[0,1]$ with density $f$. Assume there exist $C>0$ and $p>1$ such that, for all $0<r<R<1$ and $x$ in the support of $\mu$,

$$\frac{\mu(B(x,R))}{\mu(B(x,r))}\geqslant C\left(\frac{R}{r}\right)^{1-1/p}.$$

Let $\{I_i\}$ be a finite set of pairwise disjoint intervals satisfying $\mu(I_i)\geqslant\varrho\lambda(I_i)$, where $\lambda$ is Lebesgue measure. Assume additionally that $C\geqslant1$. Disjoint-interval density conjecture. Then

$$\varrho^{\,p}\lesssim\frac{1}{\sum_{i=1}^N\lambda(I_i)}.$$

This is suggested by an attempted iteration of the preceding lemma for combining separated intervals. The authors state that they were unable to prove such an iteration, and no resolution of this proposed statement is given.