The deletion-normality conjecture for non-complete graphs

For a graph $H$, an $H$-deletion-saturated graph is an $H$-free graph $G$ with $E(G)\neq\varnothing$ such that, for every $e\in E(G)$, the graph $G-e$ is not $H$-free. A graph $H$ is deletion-normal if there exists an $H$-deletion-saturated graph. Likewise, $H$ is addition-normal if there exists an $H$-free graph $G$ with $E(\overline{G})\neq\varnothing$ such that $G+e$ is not $H$-free for every $e\in E(\overline{G})$. Deletion-normality conjecture. Every non-complete graph $H$ is deletion-normal. Equivalently, every non-edgeless graph $H$ is addition-normal. Complete graphs on at least two vertices are the only graphs currently known not to be deletion-normal, while edgeless graphs on at least two vertices are the only graphs currently known not to be addition-normal; the conjecture is proved for several classes, including complete bipartite graphs with unequal part sizes, triangle-free unicyclic graphs, graphs with two leaves at distance at most three, and line graphs of trees.