Multiplicative Duffin–Schaeffer conjecture

Let $n\geq 1$ and let $\psi:\mathbb{N}\to\mathbb{R}^{\geq 0}$ be a non-negative function. Define

$${\cal D}^\times_n(\psi):=\{(x_1,\dots,x_n)\in[0,1]^n:\prod_{i=1}^n\|qx_i\|'<\psi(q)\text{ for infinitely many }q\in\mathbb{N}\}.$$

Here $\varphi(q)$ denotes Euler's phi function. Multiplicative Duffin–Schaeffer conjecture. If

$$\sum_{q=1}^{\infty}\left(\frac{\varphi(q)}{q}\right)^n\psi(q)\log^{n-1}q=\infty,$$

then $|{\cal D}^\times_n(\psi)|=1$. This is the multiplicative analogue of the Duffin–Schaeffer conjecture. The supplied status evidence identifies a counterexample, so this formulation is refuted.