Palindromic complexity conjecture for languages with symmetric order conditions

Let $L$) be a language on $k$ letters satisfying a symmetric order condition and having no connection. Its palindromic complexity is the number of palindromic factors of each length. Palindromic complexity conjecture. The palindromic complexity of $L$ is $1$ for even $n$ and $k$ for odd $n$. The conjecture is motivated by examples and by results suggesting that connections occurring in suitably paired bispecial factors do not reduce the number of palindromes; its general validity is not established in the source.