Lyngsø and Pedersen's antipalindromic subsequence conjecture

A binary word is a word over the alphabet $\{0,1\}$; a circular word is an equivalence class of linear words under rotations; an antipalindrome is a binary word whose letters in opposite positions are distinct. A subsequence of a circular word is a subsequence of one of its linear representations.

Lyngsø and Pedersen's conjecture. Every binary circular word of length $n$ divisible by $6$ with equal numbers of zeros and ones has an antipalindromic subsequence of length at least

$$\frac{2}{3}n.$$

The conjecture arose from an approximation algorithm for a two-dimensional protein-folding problem. The source reports no progress toward a proof; the bound $\frac{1}{2}n$ is known, but no better bound is proved.