Nonexistence of odd-dimensional weakly Hadamard diagonalizable graphs with diagonalizable Laplacians

A graph $X$ is weakly Hadamard diagonalizable if its Laplacian matrix is diagonalized by a weak Hadamard matrix, meaning a matrix with pairwise orthogonal columns whose entries satisfy the weak Hadamard conditions. The Laplacian eigenspaces have equal algebraic and geometric multiplicities when the Laplacian matrix is diagonalizable.

Odd-dimensional nonexistence conjecture. For any odd dimension, there is no weakly Hadamard diagonalizable graph whose Laplacian eigenspaces have equal algebraic and geometric multiplicities; equivalently, there is no weak Hadamard matrix with pairwise orthogonal columns diagonalizing the Laplacian matrix of the graph.

This conjecture is motivated by the preceding lemma and corollary concerning weakly Hadamard diagonalizable graphs and their Laplacian eigenspaces. Its resolution would rule out diagonalizable examples in every odd dimension, but the supplied text does not state whether it is known or open.