Regular-or-biregular eigenvector conjecture for the square of the adjacency matrix

Let $G$ be a connected graph, let $A$ be its adjacency matrix, let $d_v$ denote the degree of vertex $v$, and let $D$ be the diagonal degree matrix. Thus $D^{1/2}\mathbf{1}$ is the vector whose $v$-th entry is $\sqrt{d_v}$. 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 eigenvector conjecture. If

$$D^{1/2}\mathbf{1}$$

is an eigenvector of $A^2$, then $G$ is regular or biregular. This conjecture is presented as an experimentally supported partial reformulation related to the preceding conjecture; the paper does not establish it in general, so it remains open.