Strict diameter-based eigenvalue bound for non-tree quantum graphs

Let $\mathcal{G}$ be any connected, compact graph with total length $L$ and diameter $D$, and let $\mu_k(\mathcal{G})$ denote its $k$-th eigenvalue. Assume that $L/k>D/2$. Strict diameter-based eigenvalue conjecture. Then $\mu_k(\mathcal{G})$ is strictly larger than the square of the smallest positive solution $\omega>0$ of

$$\cos\left(\frac{\omega D}{2}\right)=\omega\left(\frac{L}{k}-\frac{D}{2}\right)\sin\left(\frac{\omega D}{2}\right).$$

The conjecture proposes a sharper form of the preceding diameter-based bound, removing the cycle-rank term and the assumption excluding long loops. The cited examples of loops and tadpoles motivate the claim, while its validity for arbitrary connected compact graphs remains open.