The weak p-adic digit distribution conjecture for algebraic irrational numbers

Let $\alpha\in\overline{\mathbf{Q}}\setminus\mathbf{Q}$, and let $S$ be the set of rational primes that are totally split in $\mathbf{Q}(\alpha)$. Let $r$ and $s$ be integers, and let $u,v$ be real numbers with $0\leq u<v\leq 1$. For $p\in S$ and a field embedding $\sigma\colon\mathbf{Q}(\alpha)\to\mathbf{Q}_p$, write $\alpha_n(p,\sigma)$ for the $n$th $p$-adic digit of $\alpha$. Weak p-adic digit distribution conjecture. For all but finitely many primes $p\in S$, with the finite exceptional set depending on $\alpha$, $r$, $s$, $u$, and $v$, the digit $\alpha_n(p,\sigma)$ lies in the interval $(u(p-1),v(p-1))$ for infinitely many integers $n\geq 0$ in the arithmetic progression $n=rm+s$. This is presented as a weakened form of normality for algebraic $p$-adic numbers; the corresponding normality question for algebraic numbers of degree at least two is described as open, and no resolution of this weaker conjecture is given.