Inhomogeneous multiplicative Khintchine-type conjecture

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

$$\psi:\mathbb Z_{\geqslant 2}\to\mathbb R_{\geqslant 0},\qquad \psi(n)=n^{-1}(\log n)^{-k}.$$

Inhomogeneous multiplicative Khintchine-type conjecture. Then for almost all $\alpha_k\in\mathbb R$, there exist infinitely many $(n_1,\ldots,n_k)\in\mathbb Z^k$ such that

$$\\|n_1\alpha_1+\cdots+n_k\alpha_k\\|<\psi(n_1^+\cdots n_k^+).$$

This is posed in the paper’s discussion of the dual problem and is therefore open in the supplied source.