Weak finitarity conjecture for Pisot units

Let $\beta$ be a Pisot unit, meaning an algebraic integer greater than $1$ whose other algebraic conjugates have modulus less than $1$ and whose norm is a unit. Call $\beta$ weakly finitary if, for every $\delta>0$ and every $x\in\mathbb{Z}[\beta]_+$, there exists $f\in\operatorname{Fin}(\beta)\cap(0,\delta)$ such that $x+f\in\operatorname{Fin}(\beta)$, where $\operatorname{Fin}(\beta)$ is the set of nonnegative real numbers having finite greedy $\beta$-expansions. Weak finitarity conjecture. Every Pisot unit is weakly finitary. Weak finitarity is introduced as an arithmetic condition relevant to bijective arithmetic codings; the passage describes this conjecture as apparently very difficult and reports it as shared by most experts, with no resolution supplied.