Polynomial convexity conjecture for the domain ${\mathcal C}$

Let ${\mathcal C}$ be the domain constructed in the paper from the matrix-valued map $X(\zeta)$.

Polynomial convexity conjecture. The domain ${\mathcal C}$ is polynomially convex.

The conjecture is the missing ingredient in the paper's argument. The authors further believe that ${\mathcal C}$ is the 27-dimensional exceptional Cartan domain, which would imply polynomial convexity, but this stronger identification is not part of the stated conjecture.