Characterization of iterated palindromes in the intersection of two clustering classes

Let $A=\{a,b,c\}$ be the ternary alphabet, let $P_1$ and $P_2$ denote the two classes of perfectly clustering words, and let $\operatorname{Pal}(u)$ be the iterated palindrome directed by a word $u\in A^*$. A word $w$ belongs to $P_1\cap P_2$ exactly when there is a word $u\in\{a,c\}^*\cdot b^*$ such that

$$\operatorname{Pal}(u)=w.$$

Characterization conjecture. A word $w\in P_1\cap P_2$ if and only if $\operatorname{Pal}(u)=w$ for some $u\in\{a,c\}^*\cdot b^*$. The preceding proposition establishes this characterization for iterated palindromes, and the conjecture is presented as supported by computer exploration; no resolution is supplied here.