Weighted metric theorem for approximation by uniformly distributed sequences

Let $n\in\mathbb N$, let $\omega=(\boldsymbol{\omega}_j)_{j\geq 1}$ be a sequence in $[0,1]^n$, and let $\Psi=(\psi_1,\ldots,\psi_n)$ be an $n$-tuple of non-increasing functions. Write $W_{\omega}(\Psi)$ for the associated weighted approximation set, and let $\lambda_n$ denote $n$-dimensional Lebesgue measure. Weighted metric theorem. For almost every sequence $\omega$ in $[0,1]^n$, for every such $\Psi$ we have

$$\lambda_n(W_{\omega}(\Psi)) = \begin{cases} 0 & \textup{if } \displaystyle\sum_{j=1}^{\infty} \prod_{i=1}^n \psi_i(j) < \infty, \\ 1 & \textup{if } \displaystyle\sum_{j=1}^{\infty} \prod_{i=1}^n \psi_i(j) = \infty. \end{cases}$$

This would extend the cited non-weighted zero-one law to arbitrary coordinate-wise approximation functions. The preceding discussion notes that uniformly distributed sequences need not satisfy the divergent-case conclusion, so the conjecture concerns almost every sequence rather than every uniformly distributed sequence; its resolution is not given in the source.