Weak inhomogeneous Duffin–Schaeffer conjecture for systems of linear forms

Fix $m,n\in\mathbb N$ and an inhomogeneous parameter $\mathbf y\in\mathbb R^m$. For every $\mathbf q\in\mathbb Z^n$, define

$$P(\mathbf q)=\mathbf y+\mathbb Z^m.$$

Let $W_{n,m}^P(\psi)$ be the corresponding limsup set and let $\operatorname{m}$ denote Lebesgue measure. Weak inhomogeneous Duffin–Schaeffer conjecture. If

$$\sum_{\mathbf q\in\mathbb Z^n\setminus\{\mathbf0\}}\left(\frac{\varphi(\gcd(\mathbf q))\psi(\mathbf q)}{\gcd(\mathbf q)}\right)^m=\infty,$$

then

$$\operatorname{m}(W_{n,m}^P(\psi))=1.$$

The source explains that this is the inhomogeneous analogue of the weak dual Duffin–Schaeffer statement and that the corresponding homogeneous result follows from the Duffin–Schaeffer theorem. It does not state that the inhomogeneous version has been proved.