Asymptotic conjecture for the spectral Nordhaus–Gaddum function $f_1(n)$

Let $f_1(n)$ denote the spectral Nordhaus–Gaddum function defined in the paper for graphs of order $n$. Asymptotic conjecture for $f_1(n)$.

$$f_1(n)=\frac{4n}{3}+O(1).$$

The preceding construction gives the lower bound $f_1(n)>4n/3-2$, while the paper notes that the available upper bound is far from best; the conjecture predicts the correct asymptotic growth up to an additive constant.