Odd-order graphs do not exhibit Laplacian perfect state transfer

Let $X$ be a simple connected unweighted graph on an odd number of vertices. Odd-order Laplacian PST conjecture. Laplacian perfect state transfer does not occur in $X$. This extends the corresponding result for unweighted graphs on $3$ or $5$ vertices and follows for weighted join graphs under the additional assumption that the characteristic polynomial of the Laplacian has integer coefficients; the general case remains open.