The golden-ratio conjecture for radii of convergence of q-deformed real numbers

Let $[x]_q$ denote the $q$-deformation of a real number $x>0$, viewed as a power series in the complex variable $q$, and let $R(x)$ be its radius of convergence. For the golden ratio $\varphi=(1+\sqrt{5})/2$, write $R(\varphi)$ for the radius of convergence of $[\varphi]_q$; explicitly, $R(\varphi)=(3-\sqrt{5})/2$. Golden-ratio radius conjecture. For every real $x>0$, the radius of convergence satisfies

$$R(x)\geq R(\varphi),$$

and equality holds only for those $x$ that are $\operatorname{PSL}(2,\mathbb{Z})$-equivalent to $\varphi$. This conjecture identifies the golden ratio as the extremal case for the convergence radius of the $q$-deformation; the supplied evidence states that the claim was proved for rational $x$, while the corresponding assertion for irrational $x$ is not resolved here.