Blanco–Buehrle spectral gap conjecture for generalised pancake graphs

Let $\mathcal P_m(n)$ denote the generalised pancake graph, and let $\lambda_1(\mathcal P_m(n))$ and $\lambda_2(\mathcal P_m(n))$ be its largest and second-largest signless Laplacian eigenvalues. Blanco–Buehrle spectral gap conjecture. For an integer $m\geqslant 2$,

$$\lambda_1(\mathcal P_m(n)) - \lambda_2(\mathcal P_m(n)) \to \begin{cases} 1 & \text{if } m=2; \\ 2 & \text{if } m\geqslant 3; \end{cases}$$

as $n\to\infty$. This combines parts of conjectures of Blanco and Buehrle concerning the asymptotic spectral gap of generalised pancake graphs. The paper proves an upper bound consistent with the claim, but states that the conjecture remains open for $m\geqslant 2$.