Wavelet determination of the persistence diagram of the phase space

Let $C$ be the phase space for the universal quasiperiodic factorization, with $u=U\circ\Phi$ and $\Phi:S^1\to C$. An appropriate wavelet basis is used to decompose the signal $u$ into wavelet coefficients. Wavelet persistence conjecture. The persistence diagram of $H_1(C)$ can be deduced from the coefficients of $u$ when decomposed in an appropriate wavelet basis. This would provide a way to recover the persistent homology of the phase space from wavelet data, complementing the detection of prominent specular reflections by persistent $H_1(C)$. The source does not state whether this claim has been proved or remains open.