Conjectured equality of spectral and positive-eigenvalue Nordhaus–Gaddum maxima

Let $G$ range over graphs on $n$ vertices, let $\overline G$ be its complement, let $\mu(G)$ be the spectral radius, and let $s^+(G)$ be the sum of the squares of the positive adjacency eigenvalues. Let $f(n)$ be the piecewise function defined by

$$f(n)=0$$

for $n\equiv2\pmod3$,

$$f(n)=\frac{\sqrt{(3n-2)^2+8}-(3n-2)}6$$

for $n\equiv1\pmod3$, and

$$f(n)=\frac{\sqrt{(3n-1)^2+8}-(3n-1)}6$$

for $n\equiv0\pmod3$. Square-root positive-eigenvalue conjecture.

$$\max_{v(G)=n}\bigl(\mu(G)+\mu(\overline G)\bigr)=\max_{v(G)=n}\bigl(\sqrt{s^+(G)}+\sqrt{s^+(\overline G)}\bigr)=\frac{4n}{3}-\frac53+f(n).$$

The authors report testing this conjecture on named graphs with up to 40 vertices without finding a counterexample; it remains open in the paper.