Inhomogeneous Duffin–Schaeffer conjecture for systems of linear forms

Fix $m,n\in\mathbb N$. Let $\mathbf y\in\mathbb R^m$ and define, for $\mathbf q\in\mathbb Z^n$,

$$P(\mathbf q)=\{\mathbf p+\mathbf y:\mathbf p\in\mathbb Z^m,\ \gcd(p_i,\mathbf q)=1,\ i=1,\dots,m\}.$$

Here $W_{n,m}^P(\psi)$ is the corresponding limsup set and $\operatorname{m}$ denotes Lebesgue measure. Inhomogeneous Duffin–Schaeffer conjecture.

$$\operatorname{m}(W_{n,m}^P(\psi))=\begin{cases}0&\text{if }\displaystyle\sum_{\mathbf q\in\mathbb Z^n\setminus\{\mathbf0\}}\left(\frac{\varphi(\gcd(\mathbf q))\psi(\mathbf q)}{\gcd(\mathbf q)}\right)^m<\infty,\\1&\text{if }\displaystyle\sum_{\mathbf q\in\mathbb Z^n\setminus\{\mathbf0\}}\left(\frac{\varphi(\gcd(\mathbf q))\psi(\mathbf q)}{\gcd(\mathbf q)}\right)^m=\infty.\end{cases}$$

This is the inhomogeneous analogue of the Duffin–Schaeffer theorem for systems of linear forms. The source presents it as a conjectural version, without supplying evidence of resolution.