Conjecture on automorphisms of graph squares

Let $G$ be a graph, and write $G^2$ for its square, with $Aut(G)$ and $Aut(G^2)$ denoting their respective automorphism groups. For a connected graph, let $d$ be its diameter and $r$ its radius. Automorphism conjecture for graph squares. (i) If $G$ is a connected graph with diameter $d$ and radius $r$ such that $r<d\leqslant 2r-2$, then

$$Aut(G)=Aut(G^2).$$

(ii) If $G$ is a connected bipartite graph with radius $r>2$, then

$$Aut(G)=Aut(G^2).$$

The conjecture concerns the characterization of graphs whose automorphism group is unchanged on passing to the square. The source presents this as an open problem motivated by the belief that most graphs have this property, while noting that previous attempts at a full characterization had failed.