Inhomogeneous Littlewood-type counting conjecture for inhomogeneously badly approximable numbers

Let $\alpha\in\mathcal K$ and $\gamma,\delta\in\mathbb R$, and let

$$\operatorname{Bad}(\delta)=\{\beta\in\mathbb R:\liminf_{n\to\infty}n\|n\beta-\delta\|>0\}.$$

For the counting assertion, write

$$\#\{n\in[N]:n\|n\alpha-\gamma\|\,\|n\beta-\delta\|\leqslant 1/\log n\}\gg\log\log N.$$

Inhomogeneous Littlewood-type counting conjecture. There exists $G=G(\alpha,\gamma,\delta)\subseteq\operatorname{Bad}(\delta)$ with Hausdorff dimension

$$\mathrm{dim}_{\mathrm H}(\mathcal G)=1$$

such that every $\beta\in\mathcal G$ satisfies the counting assertion for large $N\in\mathbb N$. This is posed as a desirable strengthening involving numbers that are badly approximable in an inhomogeneous sense. The supplied text does not establish whether this conjectural statement is open or resolved.