Dual inhomogeneous Gallagher problem in higher dimensions

Let $k\geqslant2$ and let $(\alpha_1,\ldots,\alpha_{k-1})\in\mathbb R^{k-1}$. For $n\in\mathbb Z$, define $n^+=\max(|n|,2)$, and for $\mathbf n=(n_1,\ldots,n_k)\in\mathbb Z^k$ define

$$H(\mathbf n)=H(n_1,\ldots,n_k)=n_1^+\cdots n_k^+.$$

Dual Gallagher problem. For almost all $\alpha_k\in\mathbb R$, there exist infinitely many $(n_1,\ldots,n_k)\in\mathbb Z^k$ such that

$$\|n_1\alpha_1+\cdots+n_k\alpha_k\|<\frac{1}{H(\mathbf n)(\log H(\mathbf n))^k}.$$

This is posed as a problem the authors intend to address in future work, so no resolution is given in the source.