Theorem on Laplacian H-eigenvalues of a k-uniform loose path of length three

Let $k$ be a positive integer, and let the $k$-uniform loose path of length $3$ be the hypergraph considered in Theorem. The conjecture. Theorem is true for the $k$-uniform loose path of length $3$. The conjecture concerns the complete determination of the Laplacian H-eigenvalues of this hypergraph; the supplied text does not indicate whether it has been resolved.