Sprindžuk's higher-dimensional Duffin–Schaeffer conjecture

Let $k\in\mathbb{N}$ and let $\psi:\mathbb{N}\to\mathbb{R}_{\geq 0}$. Define $\mathcal{A}_k”(\psi)\subset[0,1]^k$ to be the set of points $x$ for which $\\|qx-\mathbf p\\|<\psi(q)$ for infinitely many $(\mathbf p,q)=(p_1,\dots,p_k,q)\in\mathbb{Z}^k\times\mathbb{N}$ with $\gcd(p_i,q)=1$ for every $1\leq i\leq k$, where $\\|\cdot\\|$ is the supremum norm. Let $\varphi$ be Euler's totient function. Sprindžuk's higher-dimensional Duffin–Schaeffer conjecture. For every approximating function $\psi:\mathbb{N}\to\mathbb{R}_{\geq 0}$,

$$\mathcal{L}(\mathcal{A}_k”(\psi))=\begin{cases}0&\text{if }\displaystyle\sum_{q=1}^{\infty}\left(\frac{\varphi(q)\psi(q)}q\right)^k<\infty,\\\\[2ex]1&\text{if }\displaystyle\sum_{q=1}^{\infty}\left(\frac{\varphi(q)\psi(q)}q\right)^k=\infty. \end{cases}$$

For $k\geq2$ this was verified by Pollington and Vaughan, while the case $k=1$ is the Duffin–Schaeffer conjecture; together these results give a full resolution.