The bounded-family conjecture for Hadamard diagonalizable graphs

A graph $G$ is Hadamard diagonalizable if its Laplacian matrix has a full set of orthogonal eigenvectors whose entries are all in $\{\pm1\}$; the matrix formed by these eigenvectors is then a Hadamard matrix. For a positive integer $k$, consider orders $n=16k+8$.

The bounded-family conjecture for Hadamard diagonalizable graphs. For $n=16k+8$, $k=1,2,\ldots$, there are at most $26$ distinct graphs which are Hadamard diagonalizable for some Hadamard matrix of order $n$.

This conjecture is suggested by computations for orders $24$, $40$, and $56$, where the observed families contain at most $26$ graphs. The supplied text does not report a proof or disproof, so the assertion remains open here.