Conjecture on limit cycles for adjacent even cycle lengths

Let $n\ge4$ be even. Let $\rho$ and $\phi$ be the game parameters, let $\pi_0$ and $b_{n-2}$ be the quantities defined for the stationary dynamics, and let $G_{n,n-2}$, $E_{n-2}$, $E_{n,n-2}$, $H_{n,n-2}$, and $G_{n+2,n}$ denote the functions whose zero curves partition the parameter plane. For a finite word $w$, let $\Delta_w$ denote the corresponding set of initial states, and let $[1,n,1,n-2]$, $[1,n-2]$, and $[1,n]$ denote the indicated limit cycles. Adjacent-cycle conjecture. Let $n\ge4$ be even. The curve $b_{n-2}-\pi_0=0$ lies below $H_{n,n-2}=0$ and above $G_{n+2,n}=0$, with $b_{n-2}-\pi_0$ positive above and negative below the curve. If $G_{n,n-2}<0$ and $E_{n-2}\ge0$, there are precisely two limit cycles, of forms $[1,n,1,n-2]$ and $[1,n-2]$, together with an unstable equilibrium. If $E_{n-2}<0$ and $E_{n,n-2}\ge0$, there is a unique asymptotically stable limit cycle of form $[1,n,1,n-2]$, together with an unstable equilibrium. If $E_{n,n-2}<0$ and $H_{n,n-2}\ge0$, there are precisely two limit cycles, of forms $[1,n,1,n-2]$ and $[1,n]$, together with an unstable equilibrium. If $H_{n,n-2}<0$ and $b_{n-2}-\pi_0\ge0$, there is a unique asymptotically stable limit cycle of form $[1,n]$, together with an unstable equilibrium. If $b_{n-2}-\pi_0<0$ and $G_{n+2,n}\ge0$, there is a unique asymptotically stable limit cycle of form $[1,n]$, together with an unstable equilibrium. In each case, the corresponding decomposition of $\Delta_A$ in the conjecture is asserted, and every initial state in $\Delta_B-\Delta_{\overline{B}}$ eventually enters $\Delta_A$. These claims extend the proposed classification of greedy collective Parrondo dynamics beyond the regions covered by the preceding theorems; the supplied text presents them as conjectural rather than proved.