Catlin's conjecture on rational approximations

Let $\Delta_1,\Delta_2,\dots\geqslant0$, define

$$\Delta_q'=\sup_{m\geqslant1}\Delta_{qm},$$

and let $CA$ be the set of $x\in[0,1]$ for which $|x-a/q|<\Delta_q$ holds for infinitely many pairs $(a,q)\in\mathbb{Z}\times\mathbb{N}$. Let $\varphi$ be Euler's totient function. Catlin's conjecture. If

$$\sum_{q=1}^\infty \varphi(q)\Delta_q'<\infty,$$

then $\operatorname{meas}(CA)=0$, while if

$$\sum_{q=1}^\infty \varphi(q)\Delta_q'=\infty,$$

then $\operatorname{meas}(CA)=1$. Catlin's conjecture gives the expected generalization of Khinchin's theorem when non-reduced fractions are allowed. The source presents it as a conjecture following the Duffin–Schaeffer conjecture.