My Conjecture: Let $\Gamma$ be a finite, connected, undirected graph embedded on an orientable surface $S$. Let $\{ T_{\gamma_1}, T_{\gamma_2}, \dots, T_{\gamma_k} \}$be a set of $\Delta$-actions along simple closed curves $\gamma_1, \gamma_2, \dots, \gamma_k \subset S$ such that each twist $T_{\gamma_i}$ preserves the equivalence classes of primitive cycles in $\Gamma$. Then the Ihara zeta function $Z(\Gamma; u)$ remains invariant under the composition of these twists:
$$
Z\left((T_{\gamma_1} \circ T_{\gamma_2} \circ \dots \circ T_{\gamma_k})(\Gamma); u\right) = Z(\Gamma; u).