Lejeune–Rigo–Rosenfeld conjecture on Arnoux–Rauzy words

Let $w$ be an Arnoux–Rauzy infinite word. Its $2$-binomial complexity and subword complexity are functions assigning to each positive integer the number of equivalence classes of factors of that length under, respectively, $2$-binomial equivalence and equality. Lejeune–Rigo–Rosenfeld conjecture. The $2$-binomial complexity of $w$ coincides with its subword complexity. This generalizes the corresponding property known for Sturmian words; apart from the Sturmian and eventually constant cases, the Tribonacci word is the principal known example, and the conjecture remains open in general.