Convex-domain boundary rigidity conjecture

Let $\Omega \subset \mathbb{C}^d$ be a bounded convex domain and let $\xi_0\in\partial\Omega$. A holomorphic self-map is a holomorphic map $f:\Omega\to\Omega$. The tangent cone of $\overline{\Omega}$ at $\xi_0$ determines the proposed order $m=m(\xi_0)$. Convex-domain boundary rigidity conjecture. The number $m=m(\xi_0)$ depends only on the tangent cone of $\overline{\Omega}$ at $\xi_0$, and every holomorphic self-map satisfying

$$f(z)=z+o\left(\left\|z-\xi_0\right\|^m\right)$$

must be the identity: $f=\operatorname{id}$. This would extend the paper’s boundary Schwarz lemma for bounded convex domains with $C^2$ boundary to arbitrary bounded convex domains and clarify how the tangent-cone geometry controls the optimal error term; the conjecture is presented without a resolution.