2-EXPSPACE-completeness conjecture for Ramsey-number avoidance games

Let $K_n$ be the complete graph on $n$ vertices, let $\mathrm{Ramsey}(n,n)$ be the symmetric binary Ramsey number, and let $K_{\mathrm{Ramsey}(n,n)}$ be the complete graph on that many vertices. Explicit-Ramsey-number avoidance conjecture. The graph Ramsey avoidance games played on $(K_{\mathrm{Ramsey}(n,n)},K_n,\{\},\{\})$ are $2$-${\bf EXPSPACE}$-complete. This conjecture is presented as a consequence of the preceding conjectures: computing explicit Ramsey numbers and manipulating graphs of their size suggests doubly exponential space requirements under succinct input representations.