Equidistribution conjecture for Ulam words modulo integers

Let $\mathscr{U}_n$ be the set of Ulam words of length $n$, and let $\pi$ interpret a binary word as an integer. For an integer $N>1$ and a residue class $a\in\mathbb{Z}/N\mathbb{Z}$, define the relative density

$$\rho_{a,N}(n):=\frac{\#\{w\in\mathscr{U}_n\mid \pi(w)=a\pmod N\}}{\#\mathscr{U}_n}.$$

Equidistribution conjecture. For every integer $N>1$ and every $a\in\mathbb{Z}/N\mathbb{Z}$,

$$\lim_{n\to\infty}\rho_{a,N}(n)=\frac1N.$$

The claim predicts uniform distribution of Ulam words among residue classes modulo every integer. The source presents it as a conjecture motivated by numerical evidence; no proof or disproof is given.