Weak Duffin–Schaeffer conjecture

Let $\Psi:\mathbb{N}\to\mathbb{R}_{\geqslant 0}$ and define $W(\Psi)$ as the set of numbers in $[0,1]$ approximable by $a/n$ within $\Psi(n)$ for infinitely many $n\in\mathbb{N}$ and $a\in[n]$. Write $\varphi(n)$ for Euler's totient function. Weak Duffin–Schaeffer conjecture. For any $\Psi:\mathbb{N}\to\mathbb{R}_{\geqslant 0}$ such that

$$\sum_{n=1}^{\infty}\varphi(n)\Psi(n)=\infty,$$

we have

$$\lambda\bigl(W(\Psi)\bigr)=1.$$

This is a weaker form of the Duffin–Schaeffer conjecture, stated as an open problem in the paper's discussion.