PSPACE-completeness conjecture for unrestricted graph Ramsey games

Let $G$ be a graph and let $A$ be the achievement graph, with no precolored red or green edges, denoted by $(G,A,\{\},\{\})$. Unrestricted graph Ramsey-game conjecture. Graph Ramsey games played on $(G,A,\{\},\{\})$ are ${\bf PSPACE}$-complete. The conjecture concerns the complexity of deciding whether the first player has a forced win from the uncolored graph, which is not implied by PSPACE-completeness for arbitrary positions; tractable subcases provide contrary evidence for some restricted families.