The q-rational radius-of-convergence conjecture

For a real number $x>0$, let $[x]_q$ denote the associated $q$-deformation and let $R(x)$ be the radius of convergence of its Taylor series. Write

$$\varphi=\frac{1+\sqrt{5}}{2}$$

for the golden ratio, and let $\operatorname{PSL}(2,\mathbb{Z})$ act on positive real numbers by fractional linear transformations. The q-rational radius-of-convergence conjecture. For every real $x>0$,

$$R(x)\geq R(\varphi)=\frac{3-\sqrt{5}}{2},$$

and equality holds only when $x$ is $\operatorname{PSL}(2,\mathbb{Z})$-equivalent to $\varphi$. The conjecture would imply that the singular set governing specialized Burau representations lies in the annulus $\frac{3-\sqrt{5}}{2}\leq |q|\leq \frac{3+\sqrt{5}}{2}$, yielding faithfulness outside the corresponding annulus. It has been proved in some special cases.