Conjecture on convergence of approximating densities for Bernoulli convolutions

Let $\nu_{\beta}$ be the Bernoulli convolution associated with $\beta$, and let $f_n$ denote the sequence of approximating functions introduced in the paper. Convergence conjecture. If $\nu_{\beta}$ is singular, then

$$\lim_{n\to\infty} f_n(x)=0$$

almost everywhere. If $\nu_{\beta}$ is absolutely continuous, then $\lim_{n\to\infty}f_n(x)$ exists almost everywhere and is equal to the density of $\nu_{\beta}$. This asks whether absolute continuity of $\nu_{\beta}$ is characterized by almost-everywhere convergence of the approximating densities, with the singular case corresponding to convergence to zero.