Sufficient conditions for Nash equilibria in non-zero-sum Dynkin games

Let $X^1,X^2,Y^1,Y^2,Z^1,Z^2$ be the payoff processes of a non-zero-sum Dynkin game, with payoff functions

$$R^1(\tau,\sigma)=\mathds{1}_{\{\tau<\sigma\}}X^1_\tau+\mathds{1}_{\{\sigma<\tau\}}Y^1_\sigma+\mathds{1}_{\{\sigma=\tau\}}Z^1_\tau,$$ $$R^2(\tau,\sigma)=\mathds{1}_{\{\sigma<\tau\}}X^2_\sigma+\mathds{1}_{\{\tau<\sigma\}}Y^2_\tau+\mathds{1}_{\{\sigma=\tau\}}Z^2_\sigma.$$

Define the regions $\mathbb{Y}^0_t,\mathbb{Y}^1_t,\mathbb{Y}^2_t,\mathbb{Y}^3_t$ as in the source, define $\rho$ as the first time either $X^1$ or $X^2$ reaches its corresponding $Y^i$, and define $\widehat\tau$ and $\widetilde\sigma$ as the first times after $\tau$ and $\sigma$ when $Z$ leaves $\mathbb{Y}^1$ and $\mathbb{Y}^2$, respectively. Non-zero-sum Dynkin-game conjecture. The four conditions listed in the source—$Z_T\in\mathbb{Y}^0_T\cup\mathbb{Y}^3_T$; left upper semi-continuity and the limiting conditions on $X^1,X^2$; the stated monotonicity conditions on $X^1$; and the stated post-$\rho$ conditions on $X^2$ or $X^1$ according to whether $X_\rho$ lies in $\mathbb{Y}^1_\rho$ or $\mathbb{Y}^2_\rho$—are sufficient for existence of a Nash equilibrium. The result would extend the preceding special case, for which the paper states a theorem guaranteeing a Nash equilibrium, but no resolution of this conjecture is supplied.