$C^1$-rigidity conjecture at critical points

Let $f$ and $\tilde f$ be maps in the setting of the main theorems, and let $c$ be a critical point. Write $\omega(c)$ for the omega-limit set of $c$, and call it of bounded geometry when the bounded-geometry condition from the main theorems holds. $C^1$-rigidity conjecture. Whenever $\omega(c)$ is minimal and has bounded geometry, the conjugacy is differentiable at each critical point $c$. This is presented as a weaker form of the $C^1$-rigidity available for bounded-type infinitely renormalizable maps; the general assertion is left as a belief rather than established in the paper.