The fundamental-period dichotomy for $v$-palindromes

Let $n$ be as in Theorem. The quantities $\omega_0(n)$ and $\omega_f(n)$ denote, respectively, the minimal period and a constructed period of $n$.

Fundamental-period conjecture. Either

$$\omega_0(n)=1$$

or

$$\omega_0(n)=\omega_f(n).$$

This conjecture was proposed after computing the two periods for small values of $n$; the supplied text gives no resolution beyond the later discussion of counterexamples to this and a refinement.