The dimension bounds conjecture for restricted numerator systems

Let $B_n\subseteq\{0,1,\dots,n-1\}$ for each integer $n$. For an approximation function $f$ and an inhomogeneous shift $\theta$, define $W_B(f,\theta)$ as the set of $x\in[0,1]$ satisfying

$$\left|x-\frac{m+\theta(n)}{n}\right|\leq\frac{f(n)}{n}$$

for infinitely many pairs $n,m$ with $m\in B_n$. Let $W_0(f,\mathbf{0})$ denote the unrestricted homogeneous set. The dimension bounds conjecture. For every approximation function $f$ and inhomogeneous shift $\theta$,

$$\liminf_{n\to\infty}\frac{\log|B_n|}{\log n}\dim_H W_0(f,\mathbf{0})\leq\dim_H W_B(f,\theta)\leq\limsup_{n\to\infty}\frac{\log|B_n|}{\log n}\dim_H W_0(f,\mathbf{0}).$$

In particular, if

$$\lim_{n\to\infty}\frac{\log|B_n|}{\log n}=1,$$

then

$$\dim_H W_0(f,\mathbf{0})=\dim_H W_B(f,\theta).$$

This proposes quantitative control of the dimension loss caused by restricting the allowed numerators. The source gives no resolution.