Bi-SP Conjecture on common shortest paths in positive two-person games

Let $G=(V,E)$ be a finite two-person shortest path game with initial position $s$, terminal set $V_T$, and positive local costs $r_i(e)>0$ for $i\bin\{1,2\}$ and $e\in E$. For a stationary strategy $\sigma_1$ of player 1, let $G(\sigma_1)$ be the digraph obtained by retaining only the arc selected by $\sigma_1$ at each player-1 position. Let $P_2(\sigma_1)$ be the set of paths from $s$ to $V_T$ minimizing player 2's effective cost in $G(\sigma_1)$, and let $P_2=\bigcup_{\sigma_1\in S_1}P_2(\sigma_1)$. Define $P_1$ symmetrically. An $(s,V_T)$-path is a directed path from $s$ to a vertex in $V_T$.

Bi-SP Conjecture. If $G$ has an $(s,V_T)$-path, then $P_1$ and $P_2$ have such a path in common, that is,

$$(P_1\cap P_2)\setminus\{\overline\infty\}\neq\emptyset.$$

This is the graph-theoretic reformulation of the conjectured existence of a terminal Nash equilibrium. The paper proves that $P_1\cap P_2$ is always nonempty, but the common element may be the infinite-cost symbol $\overline\infty$; the asserted terminal-path conclusion remains open.