Strong isomorphism conjecture for immersed regions in two dimensions

Let $\Omega$ and $\Omega'$ be immersed regions in the sphere $\mathbf{S}^2$, and 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 is presented as a stronger consequence of the Abelian-sector conjecture for the figure-8 region and would make the information convex set depend only on the homeomorphism type of the immersed region. Its general validity remains open.