Periodicity conjecture for Hankel determinants of extremal Dyck-path avoiding sets

Let $m\geq 3$ be an integer, let $V=\{1,2,\dots,m-1\}$, and let $D^{(m,V)}$ denote the generating series for Dyck paths whose peaks avoid the heights in the congruence classes specified by $(m,V)$. Write $H(D^{(m,V)})$ for the sequence of Hankel determinants of this series.

Extremal-set periodicity conjecture. The sequence $H(D^{(m,V)})$ is periodic and has the form

$$H\big(D^{(m,V)}\big)=\begin{cases} (1,\underbrace{0,\dots,0}_{m-2},1,1)^* & \text{if }m\equiv 1,2\pmod{4},\\ (1,\underbrace{0,\dots,0}_{m-2},-1,-1,-1,\underbrace{0,\dots,0}_{m-2},1,1)^* & \text{if }m\equiv 0,3\pmod{4}. \end{cases}$$

This gives an explicit periodic sequence for the extremal avoiding set. The paper presents this as part of its investigation of periodicity; the concluding remarks state that many periodic cases remain outside the sufficient condition and pose broader questions about characterizing all pairs $(m,V)$ with periodic $H(D^{(m,V)})$.