Conjectures on when the generalized rotor-router sequence is Sturmian

Fix positive integers $r$ and $s$, and let $w_{r,s}$ be the binary word associated with the generalized one-dimensional rotor-router model. The source also gives the parameters

$$\alpha=\frac{\sqrt{s}}{\sqrt{r}+\sqrt{s}},\qquad \beta=\frac{\alpha-1}{r}+\frac12.$$

Sturmian-region conjectures. (i) If $-4\leq r-s\leq3$, then, except for $(r,s)=(4,1)$, $w_{r,s}$ is Sturmian. (ii) If $r-s=4$, then $w_{r,s}$ is Sturmian if and only if $r$ is even. (iii) If $r-s<-4$, then, with finitely many exceptions, $w_{r,s}$ is not Sturmian. These conjectures concern the observed diagonal stripe of Sturmian parameters; the source gives no resolution.