Let $ϵ,\delta >0$ and $n$ be sufficiently large in terms of $ϵ$ and $\delta$. Let $G$ be a triangle-free graph on $n$ vertices with maximum degree $<n^{1/2-ϵ}$. Can $G$ be made into a triangle-free graph with diameter $2$ by adding at most $\delta n^2$ edges?