The strict upper-bound conjecture for graph Ky Fan norms

Let $xi_k$ denote the asymptotic normalized maximum of the Ky Fan $k$-norm over graphs, as defined in the paper.

Strict upper-bound conjecture. There exist infinitely many positive integers $k$ such that

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

The preceding results establish equality for square values $k=s^2$ under the stated regular-matrix condition; the conjecture concerns infinitely many nonsquare cases where the general upper bound is not attained.