Regular-or-biregular conjecture for powers of the adjacency matrix and normalized Laplacian

Let $G$ be a connected graph, let $A$ be its adjacency matrix, and let $\mathcal{L}$ be its normalized Laplacian. A graph is regular if all its vertices have the same degree, and biregular if its vertices have two degree values such that every edge joins vertices of different degree values. Regular-or-biregular conjecture. If, for some polynomial $f$ and integer $r>0$,

$$A^r=f(\mathcal{L}),$$

then $G$ is regular or biregular. The conjecture is motivated by the lack of a complete characterization of the relation $A^r=f(\mathcal{L})$; the paper proves the conclusion when $r$ is odd, but the general case remains open.