A zero-one law for improvements to Dirichlet's theorem in arbitrary dimension

Let $ψ$ be a continuous decreasing function defined for all sufficiently large positive real numbers. We say that a real $m\times n$ matrix $A$ is $ψ$-Dirichlet if for every sufficiently large real number $t$, one can find $\mathbf{p} \in \mathbb{Z}^m$, $\mathbf{q} \in \mathbb{Z}^n\setminus\{\mathbf