Finite–infinite separation conjecture for interval discrepancy

Let $\mathcal{S}=(\mathcal{I}_1,\mathcal{I}_2,\dots)$ be an infinite-length strategy, and define

$$\operatorname{disc}_n(\mathcal{S})=\max_{t\leq n}\operatorname{disc}(\mathcal{I}_t).$$

Here $\operatorname{disc}(n)$ is the optimal discrepancy for finite strategies of length $n$. Finite–infinite separation conjecture. For any infinite-length strategy $\mathcal{S}$,

$$\liminf_{n\to\infty}\frac{\operatorname{disc}_n(\mathcal{S})-\operatorname{disc}(n)}{1/n}>0.$$

If true, this would establish a positive asymptotic separation between infinite-length strategies and optimal finite-length strategies. The paper presents this as a weaker conjecture than the preceding open question about the sharper lower bound $2-3/(2n)-O(1/n^2)$, and it remains open.