Non-localization conjecture for discrete-time quantum walks with continuous spectrum

Let $WC$ be the quantum walk operator of a discrete-time quantum walk (DTQW), and let its distribution be the associated position distribution. The operator has only continuous spectra when its spectrum contains no point-spectrum component, and it has no embedded eigenvalues when no eigenvalue lies inside the continuous spectrum. Non-localization conjecture. The distribution of the DTQW is not localized if the quantum walk operator $WC$ only has continuous spectra and does not have embedded eigenvalues. The conjecture proposes a general relation between the spectral type of the quantum walk operator and localization; the source presents it as unresolved, alongside open questions about recurrence and rational-parameter limit distributions.