Exact finite interval discrepancy conjecture

For each positive integer $n$, let $\operatorname{disc}(n)$ be the minimum, over all strategies of length $n$ that repeatedly split an existing interval into two starting from $[0,1]$, of the maximum ratio between the longest and shortest interval at any intermediate stage. Exact discrepancy conjecture. For every positive integer $n$,

$$\operatorname{disc}(n)=2^{1-1/\lceil n/2\rceil}.$$

The equality would show that the lex-merge construction is optimal. The paper establishes the upper bound given by the right-hand side and a lower bound of $2^{1-1/\lceil n/3\rceil}$, so the conjecture remains open.