Brevier–Preissmann–Sebő strong antipalindromic subsequence conjecture

Let $w$ be a binary circular word of length $n$ divisible by $6$ with equal numbers of zeros and ones. A circular word can be represented as a concatenation of linear words, and a subsequence is taken in the corresponding linear representation. An antipalindrome is a binary word whose opposite letters are distinct.

Brevier–Preissmann–Sebő's conjecture. The word $w$ can be partitioned into two linear words $w_1,w_2$ of equal length, $w=w_1w_2$, having subsequences $s_1,s_2$ such that $s_1s_2$ is an antipalindrome and

$$|s_1|=|s_2|=\frac{1}{3}|w|.$$

This strengthens the Lyngsø–Pedersen conjecture by requiring the two halves of the antipalindromic subsequence to come from opposite halves of a balanced cut of the circle. It was checked computationally through $n=30$; the source gives a weaker proved bound with each half of length $\frac14|w|$.