Bounded-average-partial-quotient approximation conjecture

Fix an integer $B\geq2$, and let $\mathcal{C}_B$ be the set of irrational numbers whose partial quotients are bounded in average by $B$. For each $n$, consider rational points $k/n$ with $k\in[n]$. Bounded-average approximation conjecture. For some integer $B\geq2$,

$$\inf_{\alpha\in\mathcal{C}_B}\min_{k\in[n]}\left|\frac{k}{n}-\alpha\right|\ll\frac{\log n}{n^2}.$$

The paper cannot prove this approximation statement, although it believes it holds for some very small $B$; the preceding proposition shows that it would imply the weak logarithmic discrepancy conjecture.