Existence conjecture for doubly saturated Ramsey-good graphs

Let $s,t \ge 3$ with $s \le t$ and $(s,t) \notin \{(3,4),(3,6)\}$. A graph is doubly saturated $R(s,t)$-good if it is $R(s,t)$-good and remains so under the relevant double-saturation condition described in the paper. Existence conjecture. For every such pair $(s,t)$, there is a doubly saturated $R(s,t)$-good graph. The conjecture extends the authors' construction of infinitely many circulant examples, while the exceptional parameter pairs are excluded; its general status is open.