On automorphisms of distance-regular graphs with intersection arrays $\{2r+1,2r-2,1;1,2,2r+1\}$

Let $\Gamma$ be an antipodal graph with intersection array $\{2r+1,2r-2,1;1,2,2r+1\}$, where $2r(r+1)\le 4096$. If $2r+1$ is a prime power, then Mathon's scheme provides the existence of an edge-symmetric graph with this intersection array. Note that $2r+1$ is not a prime power only for $r\in \{7,17,19,22,25,27,31,32,37,38,42,43\}$. We study automorphisms of hypothetical distance-regular graphs with the specified values of $r$. The cases $r\in \{7,17,19\}$ were considered earlier. We prove that, if $\Gamma$ is a vertex-symmetric graph with intersection array $\{2r+1,2r-2,1;1,2,2r+1\}$, $2r+1$ is not a prime power, and $r\le 43$, then $r=25,27,31$.