The rigidity conjecture for Hausdorff dimensions under inhomogeneous shifts

Let $B_n\subseteq\{0,1,\dots,n-1\}$ for each integer $n$. For an approximation function $f$ and an inhomogeneous shift $\theta$, define

$$W_B(f,\theta)=\left\{x\in[0,1]:\left|x-\frac{m+\theta(n)}{n}\right|\leq\frac{f(n)}{n}\text{ for infinitely many pairs }n,m\text{ with }m\in B_n\right\}.$$

Write $\mathbf{0}$ for the zero shift. The rigidity conjecture. For every approximation function $f$ and inhomogeneous shift $\theta$,

$$\dim_H W_B(f,\theta)=\dim_H W_B(f,\mathbf{0}).$$

This conjecture asks whether the Hausdorff dimension is invariant under inhomogeneous shifts even for general systems of allowed numerators $B_n$. The source gives no resolution.