A strict asymptotic bound conjecture for Ky Fan graph eigenvalue sums

For a graph $G$ of order $n$, let $\lambda_1^*(G)\geq\lambda_2^*(G)\geq\cdots$ be the eigenvalues of its adjacency matrix in nonincreasing order of absolute value, and let $\xi_k(n)$ be the maximum of $\lambda_1^*(G)+\cdots+\lambda_k^*(G)$ over graphs of order $n$. Define

$$\xi_k=\lim_{n\rightarrow\infty}\frac{\xi_k(n)}{n}.$$

Strict asymptotic bound conjecture. There exist infinitely many integers $k$ such that

$$\xi_k<\frac{1+\sqrt{k}}{2}.$$

The upper bound is attained asymptotically when $k$ is a square under the matrix condition discussed in the source, while the conjecture concerns infinitely many nonsquare cases where a strict inequality should hold.