The equivalence conjecture for Hadamard diagonalizable graphs

Let $H_1$ and $H_2$ be equivalent normalized Hadamard matrices, meaning that one can be obtained from the other by negating rows or columns and permuting rows or columns, while remaining normalized. A graph $G$ is Hadamard diagonalizable by $H$ if the columns of $H$ are eigenvectors of the Laplacian matrix of $G$.

The equivalence conjecture for Hadamard diagonalizable graphs. If $H_1$ and $H_2$ are equivalent normalized Hadamard matrices, then $G$ is Hadamard diagonalizable by $H_1$ if and only if $G$, up to some relabeling, is Hadamard diagonalizable by $H_2$.

The paper motivates this as a possible invariance of the associated graph family under equivalence of normalized Hadamard matrices. The supplied text does not state whether this claim has been proved or disproved.