PSPACE-completeness conjecture for symmetric binary graph Ramsey avoidance games

Let $K_k$ and $K_n$ be complete graphs, let $E^r$ and $E^g$ be precolored red and green edge sets, and let $\mathrm{Ramsey}(n,n)$ denote the classic symmetric binary Ramsey number. Classic Ramsey-number avoidance conjecture. Graph Ramsey avoidance games played on $(K_k,K_n,E^r,E^g)$ where $k\geq\mathrm{Ramsey}(n,n)$ are ${\bf PSPACE}$-complete. The conjecture restricts avoidance games to complete host graphs whose order is at least the corresponding symmetric binary Ramsey number; the paper presents these games as difficult, but supplies no resolution of the conjecture.