The standard-flow simplex conjecture for symmetric convex ancient solutions

Let $X_s$ be the space of point-symmetric convex hypersurfaces, let $I(h_0,h_1)$ be the invariant set cut out by the Huisken-energy bounds $h_0$ and $h_1$, and let $I_s(h_0,h_1)$ denote its point-symmetric part. Write

$$\Delta_n=\left\{(a_0,\ldots,a_n)\in{\mathbb R}^{n+1}\mid a_i\geq0,\ a_0+\cdots+a_n=1\right\}$$

for the standard $n$-simplex, and let $\sigma^\tau$ be the standard flow on $\Delta_{n-1}$. For $n\geq2$, suppose that $0<h_0<{\mathcal H}(\Sigma^n)$ and ${\mathcal H}(\Sigma^1)<h_1<2$.

Standard-flow simplex conjecture. The invariant set

$$I_s(h_0,h_1)/{\mathrm{SO}}_{n+1}=I(h_0,h_1)\cap X_s/{\mathrm{SO}}_{n+1}$$

is homeomorphic to an $(n-1)$-dimensional simplex, and the semiflow $\phi^t$ on this quotient is topologically conjugate to the standard flow on $\Delta_{n-1}$.

This conjecture is motivated by uniqueness results for ancient mean-curvature-flow solutions and by the paper's shadowing and dynamical results, but the supplied text gives no proof of the full homeomorphism and conjugacy.