Fourier-frame spectrality conjecture for non-overlapping Bernoulli convolutions

Let $\mu_{\rho,d}$ be the Bernoulli convolution with contraction parameter $0<\rho\le1/2$ and scale parameter $d>0$. A measure is spectral when its associated $L^2$ space admits an orthonormal basis of exponential functions. Fourier-frame spectrality conjecture. The space $L^2(\mu_{\rho,d})$ admits a Fourier frame if and only if

$$\rho=\frac{1}{2k}$$

for some $k\in\mathbb N$. Equivalently, in the non-overlapping range, the existence of an exponential Fourier frame should coincide with spectrality for Bernoulli convolutions. This conjecture extends Dai's full spectrality classification to Fourier frames for non-overlapping Bernoulli convolutions; the present method establishes nonexistence only in the odd reciprocal-power regime, leaving the remaining parameters open.