Weak palindromic subsequence conjecture for binary circular words

A binary circular word is a circular word over the alphabet $\{0,1\}$, and a palindromic subsequence reads identically from both ends.

Weak circular-palindrome conjecture. Every binary circular word of length $n$ has a palindromic subsequence of length at least

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

The source presents this as the palindromic counterpart of the antipalindromic conjectures. A bound of $\frac12n$ is immediate, and no better bound is known to be proved.