The Kurzweil-type characterization of non-singular vectors

Let $\boldsymbol{i}=(i_1,\ldots,i_n)$ be a weight vector, let $\mathcal{S}_n(\boldsymbol{i})$ denote the set of $\boldsymbol{i}$-singular vectors, and for each $\varepsilon>0$ define $\psi_\varepsilon(q)=\varepsilon q^{-1}$. Let $W_n^{\times}(\boldsymbol{i},\psi_\varepsilon)$ be the corresponding inhomogeneous weighted approximation set. Kurzweil-type characterization conjecture.

$$\bigcap_{\varepsilon>0}W_n^{\times}(\boldsymbol{i},\psi_\varepsilon)=[0,1]^n\setminus\mathcal{S}_n(\boldsymbol{i}).$$

The paper presents this as a proposed strengthening of the known implication and says that it is an interesting question in both the standard and weighted settings; a proof is not supplied.