Threshold conjecture for higher-moment star graph gaps

Let $G_{\star}$ be a star graph with $n_{\star}$ vertices, local dimension $q$, and moment $k$, and let $\Delta(G_{\star},n_{\star},k)$ denote its spectral gap. Higher-moment star-gap conjecture. Below a certain threshold for the ratio

$$\frac{k^2}{q^{n_{\star}}},$$

the star graph spectral gap satisfies

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

This is a looser conjecture intended to cover higher moments when the local dimension is sufficiently large relative to the moment. The source gives numerical motivation and notes that the unrestricted claim fails for arbitrary moments, including $k=4$ for local qubits; the precise threshold is not specified here.