Żak's graph-packing conjecture

Let $G_1$ and $G_2$ be graphs on $n$ vertices, and let $E(G_i)$ and $\Delta(G_i)$ denote the edge set and maximum degree of $G_i$, respectively. Two graphs pack if there is a bijection between their vertex sets that maps every edge of $G_1$ to a non-edge of $G_2$. Żak's graph-packing conjecture. If

$$\Delta(G_1),\Delta(G_2)\leq n-2$$

and

$$|E(G_1)|+|E(G_2)|+\max\{\Delta(G_1),\Delta(G_2)\}\leq 3n-7,$$

then $G_1$ and $G_2$ pack. Żak's preceding theorem proves the same sufficient condition with $3n-7$ replaced by $3n-96n^{3/4}-65$, so the conjecture asks for the sharp constant bound; the source provides no resolution.