Beresnevich–Haynes–Velani conjecture for coprime multiplicative approximation

Let $\psi:\operatorname{\mathbb{N}}\to[0,1/2]$ be arbitrary, let $k\geq1$, and define

$$D_k^{\times}(\psi):=\left\{\alpha\in[0,1]^k:\prod_{i=1}^k\lVert q\alpha_i\rVert'\leq\psi(q)\text{ for infinitely many }q\in\operatorname{\mathbb{N}}\right\}.$$

Here $\lVert qx\rVert':=\min_{\gcd(p,q)=1}|qx-p|$, and $\varphi$ denotes Euler's totient function.

Beresnevich–Haynes–Velani conjecture. One has

$$\lambda_k\left(D_k^{\times}(\psi)\right)=\begin{cases}1,&\text{if }\sum_{q\in\operatorname{\mathbb{N}}}\left(\frac{\varphi(q)}q\right)^k\psi(q)\log(1/\psi(q))^{k-1}=\infty,\\0,&\text{if }\sum_{q\in\operatorname{\mathbb{N}}}\left(\frac{\varphi(q)}q\right)^k\psi(q)\log(1/\psi(q))^{k-1}<\infty. \end{cases}$$

The conjecture is posed for arbitrary $\psi$; the supplied context identifies this as one of the open non-monotonic multiplicative approximation questions.