The automorphism-group characterization for token graphs of connected Cartesian products

Let $G$) be a connected graph with $n\geq 5$ vertices and prime factor decomposition

$$G=G_1\square\cdots\square G_r,$$

where $r>1$. Let $\psi$ be the homomorphism from $\operatorname{Aut}(G)$ to $\operatorname{Aut}(\mathbb{Z}_2^{[r-1]})$ defined by the induced permutation of the prime factors. Automorphism-group characterization. The automorphism group of the $k$-token graph of $G$ should satisfy

$$\operatorname{Aut}(F_k(G))\simeq \begin{cases} \mathbb{Z}_2^{[r-1]}\rtimes_{\psi}\operatorname{Aut}(G) & \textrm{if } k=2,\\ \operatorname{Aut}(G)\times\mathbb{Z}_2 & \textrm{if } k=n/2,\\ \operatorname{Aut}(G) & \textrm{otherwise.} \end{cases}$$

This gives the complete automorphism group of the token graphs in the stated range, extending the preceding lower-bound result for $F_2(G)$. The parser supplies no evidence that the characterization has been proved or disproved, so its status remains open.