The $n$-adic normal number conjecture for irrational algebraic numbers

Let $\alpha$ be an irrational algebraic number and let $n>1$ be an integer. An $n$-adic expansion of $\alpha$ is a representation $\sum_{l=0}^{\infty}a_ln^l$ with $0\le a_l<n$ satisfying the minimal polynomial equation for $\alpha$ in the $n$-adic completion $\mathbb{Z}_n$. Such an expansion is normal if, for every $m>1$ and every word $\omega\in\{0,\ldots,n-1\}^m$, the word occurs among consecutive digit blocks with limiting frequency $1/n^m$. The $n$-adic normal number conjecture. All $n$-adic expansions of $\alpha$, if they exist, are normal. This is posed as an $n$-adic analogue of the normal-number problem and remains open in the stated generality.