Ostrowski automaticity conjecture for records of $S_n(2\xi)$

For an irrational number $\xi$, define the deterministic random walk by

$$S_n(2\xi)=\sum_{j=1}^n(-1)^{\lfloor 2j\xi\rfloor}.$$

Call $r$ a record if $S_r(2\xi)$ has not occurred among the preceding partial sums, including $S_0(2\xi)=0$. A sequence is $\xi$-Ostrowski automatic if a finite-state automaton decides membership in it from the $\xi$-Ostrowski representation of an integer. Ostrowski automaticity conjecture. For every quadratic irrational $\xi$, the records of $S_n(2\xi)$ are $\xi$-Ostrowski automatic. This is presented as a variation of the Van de Lune–Arias de Reyna recurrence conjecture; the source gives explicit quadratic examples but leaves the general assertion open.