Khintchine's theorem with a moving target

Let $\gamma=(\gamma_q)_q$ be a sequence of real numbers and let $\psi:\mathbb{N}\tomathbb{R}_{\geq 0}$ be a function. Define

$$W(\psi,\gamma)=\{\alpha\in[0,1]:\|q\alpha-\gamma_q\|<\psi(q)\ \text{for infinitely many }q\inmathbb{N}\}.$$

Khintchine's moving-target conjecture. If $\psi:\mathbb{N}\tomathbb{R}_{\geq 0}$ is decreasing and $\sum_{q=1}^{\infty}\psi(q)=\infty$, then

$$m(W(\psi,\gamma))=1.$$

This asks whether the divergence part of the inhomogeneous Khintchine theorem remains valid when the target center depends on the denominator. The fixed-target theorem is classical, while the moving-target version is posed as an open problem and is not known in one dimension.