Spectral cycle-length constant conjecture

Let $C$ be a positive constant. For every $\varepsilon$ with $0<\varepsilon<C$ and all sufficiently large $n$, consider every graph $G$ of order $n$ satisfying

$$\lambda(G)>\sqrt{\left\lfloor\frac{n^2}{4}\right\rfloor}.$$

Spectral cycle-length conjecture. The largest such constant $C$ is characterized by the property that every such graph contains a cycle of every length $t\le (C-\varepsilon)n$. The source subsequently conjectures that $C=1/2$. This would improve the known bound $C=1/320$; the exact optimal constant remains open.