The largest Laplacian H-eigenvalue characterization of hyperstars

Let $k\geq 3$ be odd and let $G=(V,E)$ be a $k$-uniform connected hypergraph whose maximum degree is $d>0$. Let $\mathcal L$ be the Laplacian tensor of $G$.

Hyperstar characterization conjecture. The largest Laplacian H-eigenvalue $\lambda(\mathcal L)$ is equal to $d$ if and only if $G$ is a hyperstar.

For odd uniformity, the maximum degree is a tight lower bound for the largest Laplacian H-eigenvalue. The conjecture asserts that equality occurs exactly for hyperstars; the supplied text gives no evidence that this characterization has been resolved.