Low-moment star graph spectral-gap conjecture

For a star graph $G_{\star}$ with $n_{\star}$ vertices, let $\Delta(G_{\star},n_{\star},k)$ denote the spectral gap of its Hamiltonian at moment $k$. Low-moment star graph conjecture. The second- and third-moment star graph gaps satisfy

$$\Delta(G_{\star},n_{\star},k)>\frac{1}{2}$$

for $k=2$ and $k=3$. This conjecture would provide the spectral-gap input needed to show that local random quantum circuits form approximate designs on arbitrary connected graphs; the paper presents numerical and analytic evidence, but no proof for all $n_{\star}$.