Approximate multiplication exceptional-set conjecture

Let $r\in\mathbb{Q}$ be a nonintegral rational with $|r|>1$, and let $f_r$ be the approximate multiplication map. Define its exceptional set by

$$E(r):=\{n\in\mathbb{Z}:\text{ no iterate }f_r^{(j)}(n)\in\mathbb{Z}\text{ for }j\geq 1\}.$$

Approximate multiplication exceptional-set conjecture. The set $E(r)$ is finite. The expanding condition is necessary, and the paper proves the conjecture for rationals with denominator $2$; it remains open for rationals with denominator at least $3$.