Inhomogeneous Duffin–Schaeffer theorem

Let $\delta\in\mathbb R$, and let $\Phi:\mathbb N\to\mathbb R_{\geqslant 0}$ satisfy the two conditions denoted by

$$and$$

in the source. Inhomogeneous Duffin–Schaeffer theorem. Then for almost all $\beta\in\mathbb R$, there exist infinitely many $n\in\mathbb N$ such that

$$\\|n\beta-\delta\\|<\Phi(n).$$

The source identifies this as an open conjectural inhomogeneous analogue; it notes that the additional assumption may not be necessary and that even an analogue of Gallagher’s zero-full law is unavailable.