Uniform boundedness for rational polynomial values in multiplicative groups

Let $g(x)$ be a polynomial over the rationals of degree $d$, and let $G$ be a complex multiplicative group of rank $r$. Uniform boundedness conjecture. There is a function $b(d,r)$ such that

$$|g(\mathbb{Q})\cap G|\leq b(d,r),$$

unless $g(x)=(x+a)^\alpha$ for some rational number $a$ and integer $\alpha$.

The conjecture is presented as a consequence of the Uniformity Conjecture for rational points on curves and would make the multiplicative non-expander argument uniform in the degree and rank parameters. The source gives no resolution status.