Transitivity of the transpose group

Establish that for every transitive subgroup $G$ of $GL_n(\mathbb{F}_q)$ acting on $\mathbb{F}_q^n$, the transpose subgroup $G^T=\{A^T:A\in G\}$ is transitive on $\mathbb{F}_q^n$.

Background

The paper identifies the dual action of a transitive subgroup of $GL_n(\mathbb{F}_q)$ with the action of its transpose subgroup. It therefore isolates the transitivity assertion as a conjecture needed to complete the broader dual-layering result. The subsequent discussion emphasizes that the question is difficult and invokes the classification of transitive linear groups.