The one-dimensional Diophantine condition conjecture for inhomogeneous approximation

Let $\psi$ be a monotonic approximation function, let $\sigma_\beta$ denote the Diophantine-type function associated with an irrational number $\beta$, and let $W(\psi,\beta,\gamma,\gamma')$ be the set of pairs satisfying the associated inhomogeneous multiplicative approximation inequality for infinitely many integers $q$.

One-dimensional Diophantine condition conjecture. Suppose that

$$\sum_{q=1}^{\infty} \psi(q)\frac{\log q}{(\log\log q)^{1/2}}=\infty.$$

If $\beta$ is irrational and

$$\sigma_\beta(q)=O\bigl((\log\log q)^{1/2}\bigr),$$

then, for all real numbers $\gamma,\gamma'$, one has

$$\lvert W(\psi,\beta,\gamma,\gamma')\rvert=1.$$

The statement would reduce the joint Diophantine hypothesis on $(\gamma,\beta)$ appearing in the preceding results to a condition on $\beta$ alone. The source states it as something the authors believe, rather than as an established theorem, and gives no resolution.