Orthogonal-cut palindromic subsequence conjecture

Let $w$ be a word of length $n$ divisible by $4$, with a linear representation

$$w=w_1w_2w_3w_4$$

where the four blocks have equal length. Let $p_1p'_1$ and $p_2p'_2$ be longest palindromic subsequences with $|p_1|=|p'_1|$ and $|p_2|=|p'_2|$, where $p_1,p'_1,p_2,p'_2$ are subsequences of $w_1w_2,w_3w_4,w_2w_3,w_4w_1$, respectively.

Orthogonal-cut conjecture. For every such $w$ and every such linear representation, the maximum of the lengths of $p_1p'_1$ and $p_2p'_2$ is at least

$$\frac12n.$$

This tests the strong circular-palindrome problem using two orthogonal cuts rather than all equal-half cuts. The source offers it as a simplification motivated by the difficulty of the stronger conjecture; no resolution is reported.