The non-Liouville inhomogeneous multiplicative approximation conjecture

Let $\psi$ be an approximation function, and let $\\|\cdot\\|$ denote distance to the nearest integer. For real numbers $\beta,\gamma,\gamma'$ and $(x,y)\in[0,1]^2$, write $W(\psi,\beta,\gamma,\gamma')$ for the set of pairs satisfying the corresponding inhomogeneous multiplicative approximation inequality for infinitely many integers $q$.

Non-Liouville inhomogeneous multiplicative approximation conjecture. If

$$\sum_{q=1}^{\infty} \psi(q)\log q=\infty$$

and $\psi(q)=O\bigl(q^{-1}(\log q)^{-2}\bigr)$, then for every non-Liouville number $\gamma$ and every real number $\gamma'$, Lebesgue almost every $(x,y)\in[0,1]^2$ belongs to $W(\psi,\beta,\gamma,\gamma')$.

The source presents this as a stronger result under a non-Liouville hypothesis, while noting that the corresponding unrestricted statement is already solved. The conjecture itself remains open even under the displayed stronger decay requirement.