Three-quarter palindromic subsequence conjecture for binary words

Let $w$ be a binary word of length $n$, meaning a word over $\{0,1\}$. Assume that $w$ has no three consecutive equal letters.

Three-quarter palindrome conjecture. The word $w$ has a palindromic subsequence of length at least

$$\frac34(n-2).$$

This strengthens the proved lower bound $\frac23(n-2)$ for both palindromic and antipalindromic subsequences under the same restriction. The conjecture was checked computationally through $n=30$, with examples based on $(001)^i(011)^i$ indicating tightness.