A strict improvement for the asymptotic Ky Fan eigenvalue sum

For a graph $G$, let $\tau_k(n)$ denote the maximum, over graphs on $n$ vertices, of the sum of the $k$ largest eigenvalues of $G$, and let

$$\tau_k=\lim_{n\to\infty}\frac{\tau_k(n)}{n}$$

for fixed $k\geq 1$. The asymptotic Ky Fan eigenvalue-sum conjecture. For any $k\geq 2$, there is an $\varepsilon_k>0$ such that

$$\tau_k<\frac{1}{2}\left(1+\sqrt{k}-\varepsilon_k\right).$$

The conjecture proposes a positive improvement over the known general upper bound $\tau_k\leq \frac{1}{2}(1+\sqrt{k})$. Determining $\tau_k$ is open for every $k\geq 2$, so the conjecture remains open.