Inhomogeneous multiplicative approximation conjecture for a non-Liouville number

Let ${\alpha}, {\gamma}, {\delta} \in \mathbb R$, with ${\alpha}$ irrational and not Liouville. Let $\psi:\mathbb N\to\mathbb R_{\geqslant 0}$ be decreasing and satisfy

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

Inhomogeneous multiplicative approximation conjecture. For almost all ${\beta}\in\mathbb R$, there exist infinitely many $n\in\mathbb N$ such that

$$\|n{\alpha}-{\gamma}\|\,\|n{\beta}-{\delta}\|<\psi(n).$$

The assertion is presented as a consequence of an inhomogeneous Duffin--Schaeffer theorem and would extend the paper's proved homogeneous-fibre result to arbitrary inhomogeneous shifts. Its resolution is not supplied in the source.