Making graphs isomorphic with edge additions/removals

Consider simple graphs on the same vertex set $[n]$. For two graphs $G, H$, let $d(G, H) = min_{H’ sim H} |E(G) triangle E(H’)|$ — the smallest number of edge additions/removals needed to make $G$ isomorphic to $H$.

Let $D(G) = max_H d(G, H)$, and $S(G) = max(|E(G)|, {n choose 2} – |E(G)|)$. Clearly, $D(G) geq S(G)$, attained either on $d(G, K_n)$ or $d(G, overline{K_n})$.

Question: is $D(G) = S(G)$ for all graphs $G$? In other words, can any $G$ be made isomorphic to any $H$ in $S(G)$ edge additions/removals?

With computer search I verified this for $n leq 7$. Outside of trivial cases $H = K_n, overline{K_n}$, equality $d(G, H) = S(G)$ does not seem to hold very often: the only cases $(G, H)$ in this range of $n$ are $(K_{1, 3}, C_4)$ and $(C_5, K_{1, 4})$, and their complements. Despite this evidence, I still lack intuition for whether, or why this is always true.