Nielsen–Zhou conjecture on Schauder-basis failure of Daubechies wavelet packets

Let a Daubechies filter have length at least $4$, and let its associated basic wavelet packets be the functions obtained from the corresponding low-pass/high-pass filter pair. For $1\le p<\infty$, the packets are considered in $L^p(\mathbb R)$ as the scale varies. Nielsen–Zhou conjecture. The basic wavelet packets associated with a Daubechies filter of length at least $4$ will fail to be a Schauder basis for $L^p(\mathbb R)$ when $p\neq2$, and the functions will not be uniformly bounded in $p$-mean across scales for any $p>2$. Nielsen and Zhou established the failure of the Schauder-basis property for exponents sufficiently close to $1$ or to $\infty$ using an $\ell^1/\ell^\infty$ estimate and log-convexity; the behavior near $p=2$ remains open. The conjecture also asserts the corresponding failure of uniform $p$-mean bounds across scales for every $p>2$.