Univariate inhomogeneous Duffin–Schaeffer conjecture for systems of linear forms

Let $m,n\in\mathbb{N}$ with $n\leq 2$, fix $\mathbf{y}\in\mathbb{R}^m$, and let $\psi:\mathbb{N}\to\mathbb{R}_{\geq 0}$ be a function satisfying

$$\sum_{\mathbf{q} \in \mathbb{Z}^n} \left(\frac{\varphi(\gcd(\mathbf{q}))\psi(\lvert\mathbf{q}\rvert)}{\gcd(\mathbf{q})}\right)^m=\infty.$$

Univariate inhomogeneous Duffin–Schaeffer conjecture for systems of linear forms. For almost every $\mathbf{x}\in \operatorname{Mat}_{n\times m}(\mathbb{R})$, there exist infinitely many points $(\mathbf{p},\mathbf{q})\in\mathbb{Z}^m\times\mathbb{Z}^n$ with $\gcd(p_i,\mathbf{q})=1$ for every $i=1,\dots,m$ such that the inhomogeneous approximation condition in equation (3) holds.

This extends the stated theorem from $n>2$ to the remaining cases $n\leq 2$, completing the univariate inhomogeneous analogue of the Duffin–Schaeffer conjecture for systems of linear forms. The source gives no resolution of these cases.