Allen–Ramírez conjecture for the inhomogeneous Khintchine–Groshev theorem

For positive integers $n,m$, a function $\psi: \mathbb{N} \to \mathbb{R}_{\geq 0}$, and an inhomogeneous parameter $\mathbf{y} \in \mathbb{R}^m$, let $\mathscr{A}^\mathbf{y}_{n,m}(\psi)$ be the set of $\mathbf{x} \in [0,1)^{nm}$ such that

$$|\mathbf{qx} - \mathbf{p} - \mathbf{y}| < \psi(|\mathbf{q}|)$$

for infinitely many $(\mathbf{q}, \mathbf{p}) \in (\mathbb{Z}^n \setminus \\{\mathbf{0}\\}) \times \mathbb{Z}^m$, where $\mathbf{x}$ is an $n \times m$ matrix and $|\cdot|$ is the maximum norm. Let $|\cdot|$ also denote Lebesgue measure. Allen–Ramírez conjecture. If $(n,m)=(1,2)$ or $(2,1)$, then

$$|\mathscr{A}_{n,m}^\mathbf{y}(\psi)|=\begin{cases}0 & \text{if } \displaystyle\sum_{q=1}^{\infty}q^{n-1}\psi(q)^m<\infty,\\\\1 & \text{if } \displaystyle\sum_{q=1}^{\infty}q^{n-1}\psi(q)^m=\infty. \end{cases}$$

Thus the divergence conclusion should hold without assuming that $\psi$ is monotonic. Allen and Ramírez proved the corresponding inhomogeneous theorem for $nm\geq 3$, while monotonicity is known to be necessary in the $(1,1)$ case; the $nm=2$ cases remain open in the supplied source.