The strict inequality between the largest Laplacian H-eigenvalue and spectral radius

Let $G$ be a connected non-odd-bipartite hypergraph. Here, $\rho^{\mathcal{L}}(G)$ denotes the spectral radius of the Laplacian tensor of $G$, and $\lambda_{\max}^{\mathcal{L}}(G)$ denotes its largest H-eigenvalue.

Strict inequality conjecture.

$$\lambda_{\max}^{\mathcal{L}}(G)<\rho^{\mathcal{L}}(G).$$

The preceding results establish this inequality for the generalized power hypergraphs considered in the paper in the relevant cases, but the conjecture remains open for connected non-odd-bipartite hypergraphs in general.