A nonlinear gcd bound for products of two fractional-part approximations

Let $b\geq 2$ be the fixed integer base. Let $\alpha$ and $\beta$ be irrational algebraic numbers, and let $\epsilon>0$. For nonnegative integers $k$ and $m$, define

$$f(k,m)=[\alpha b^{k+m}][\beta b^m]+1.$$

The first nonlinear gcd conjecture. For sufficiently large $m$,

$$\operatorname{gcd}(f(k,m),b^m)<b^{\epsilon m}.$$

The source presents this as one of three illustrative open questions about extending its linear congruence estimates to nonlinear congruences.