A nonlinear gcd bound for a nested algebraic approximation

Let $b\geq 2$ be the fixed integer base. Let $\alpha$ and $\beta$ be the irrational algebraic numbers and let $\epsilon>0$ be as in the preceding conjectures. For a positive integer $m$, define

$$f(m)=[\beta[\alpha b^m]^2].$$

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

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

The source gives this as a further illustrative open question about nonlinear congruences; no resolution is supplied.