Uniform distribution of residues along powers of degree-one ideals

Let $L$ be a number field and let $\alpha\in L$ be irrational. Let $\mathfrak{a}$ be an unramified ideal of degree one that is prime to the denominator of the principal ideal $(\alpha)$. For each $l\ge 1$, write $\alpha(\mathfrak{a}^l)$ for the chosen residue representative of $\alpha$ modulo $\mathfrak{a}^l$, and let $n$ denote the norm of $\mathfrak{a}$. Uniform-distribution conjecture. The sequence

$$\left\{\frac{\alpha(\mathfrak{a}^l)}{n^l}\right\}_{l=1}^{\infty}$$

is uniformly distributed in $[0,1]$. This is presented as a reformulation of the preceding $n$-adic normality conjecture via the equivalence between normality of an $n$-adic expansion and uniform distribution of the associated residue sequence; its status is open in the source.