Rigidity of proper holomorphic self-mappings of the hexablock

The hexablock $\mathbb{H}$, introduced by Biswas-Pal-Tomar \cite{Hexablock}, is a Hartogs domain in $\mathbb{C}^4$ fibered over the tetrablock $\mathbb{E}$ in $\mathbb{C}^3$, arising in the context of $\mu$-synthesis problems. In this paper, we prove that every proper holomorphic self-map of $\mathbb{H}$ is necessarily an automorphism. Consequently, we resolve the conjecture $G(\mathbb{H}) = \mathrm{Aut}(\mathbb{H})$ on the automorphism group structure, originally posed by Biswas-Pal-Tomar in \cite{Hexablock}.