Asymptotic normality criterion for H-linear and H-circular genus distributions

Let $H$ be a graph with a self-gluing $d711: U_1 \to U_2$, and let $K$ be a valid minimal cut for this self-gluing. Construct the swapping $(H',\psi)=\operatorname{Sw}(H,d711,K)$ by cutting the edges of $K$, gluing $U_1$ to $U_2$ according to $d711$, and pairing the resulting boundary vertices by $\psi$. Let $\mathcal{G}(H,d711)$ and $\mathcal{G}^{\circ}(H,d711)$ denote the associated $H$-linear and $H$-circular families. Asymptotic normality conjecture. If the swapping $(H',\psi)=\operatorname{Sw}(H,d711,K)$ has two cycles $C_1$ and $C_2$ sharing at least one vertex, then the genus distributions of both families are asymptotically normal. The criterion is proposed as a sufficient condition because general asymptotic normality is difficult to establish when the relevant linear operator is not irreducible; the source gives no resolution of this conjecture.