Vanishing translation parameters for spectral almost-Parseval-frame towers

Let $(N_j,B_j)$ be the almost-Parseval-frame tower defined in Theorem 0.1, and let $bc$ be its associated measure. The measure $bc$ is spectral if there exists a set $bbambda\subset\mathbb{R}$ such that the exponential functions $\{e^{2\pi i\lambda x}:\lambda\inbbambda\}$ form an orthonormal basis for $L^2(bc)$. Vanishing-translation conjecture. If $bc$ is spectral, then $\alpha_j=0$ for every $j$. This conjecture concerns when the almost-Parseval-frame construction can produce a spectral measure; it predicts that spectrality forces all translation parameters in the tower to vanish. The supplied text does not state whether the conjecture has been resolved.