Strong palindromic subsequence conjecture for binary circular words

Let $w$ be a binary circular word of length $n$ divisible by $6$. Choose a linear representation and partition it as $w=w_1w_2$ into two linear words of equal length. Let $s_1$ and $s_2$ be subsequences of $w_1$ and $w_2$, respectively, and let $s_2^R$ denote reversal.

Strong circular-palindrome conjecture. There is such a partition with subsequences $s_1,s_2$ satisfying

$$s_1=s_2^R$$

—that is, $s_1s_2$ is a palindrome—and

$$|s_1s_2|=\frac{2}{3}|w|.$$

This strengthens the weak circular-palindrome conjecture by aligning the two halves of the palindrome with a cut into equal halves of the circle. The source notes that the bound $\frac12n$ is proved, while no stronger bound is known.