A gcd bound for values of two algebraic polynomials at powers of the base

Let $b\geq 2$ be the fixed integer base. Let $\epsilon>0$, let $k,l$ be positive integers, and let

$$f(x)=a_kx^k+\cdots+a_1x+a_0,$$ $$g(x)=c_lx^l+\cdots+c_1x+c_0,$$

be polynomials with real algebraic coefficients, where $a_k$ and $c_l$ are linearly independent over $\mathbb{Q}$. The gcd conjecture. For all sufficiently large $n\in\mathbb{N}$,

$$\operatorname{gcd}([f(b^n)],[g(b^n)])<b^{\epsilon n}.$$

This is proposed as a generalization of a preceding theorem concerning one polynomial and remains an open problem in the source.