Ergodic-theoretic properties of certain Bernoulli convolutions

In [17] the author and A. Vershik have shown that for $\be=\frac12(1+\sqrt5)$ and the alphabet $\{0,1\}$ the infinite Bernoulli convolution ($=$ the Erd\"os measure) has a property similar to the Lebesgue measure. Namely, it is quasi-invariant of type $\mathrm{II}_1$ under the $\be$-shift, and the natural extension of the $\be$-shift provided with the measure equivalent to the Erd\"os measure, is Bernoulli. In this note we extend this result to all Pisot parameters $\be$ (modulo some general arithmetic conjecture) and an arbitrary "sufficient" alphabet.