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

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

$$f_n(n)=\frac{\sqrt{2}n}{2}+O(1).$$

The paper proves the lower bound $f_n(n)>\sqrt{2}n/2-3$ and establishes the corresponding upper bound for $f_2(n)$, but leaves the asserted asymptotic formula for $f_n(n)$ as a conjecture.