Strong isomorphism conjecture for immersed regions in arbitrary dimensions

Let $\Omega$ and $\Omega'$ be immersed regions in the $n$-sphere $\mathbf{S}^n$, with axioms ${\bf A0}$ and ${\bf A1}$ satisfied everywhere on the reference state. Let $\Sigma(\Omega)$ denote the associated information convex set. Strong isomorphism conjecture. If $\Omega$ and $\Omega'$ are homeomorphic as immersed regions, then

$$\Sigma(\Omega)\cong\Sigma(\Omega').$$

This generalizes the two-dimensional strong isomorphism conjecture. The source presents the arbitrary-dimensional statement as a conjecture and does not establish it.