Denjoy–Wolff characterization by sequential horospheres

Let $\Omega\subset\mathbb C^N$ be a bounded convex domain whose boundary contains no non-trivial analytic discs. A sequential horosphere is a set of the form

$$E_{z_0}(\{z_n\},R)=\left\{z\in\Omega:\limsup_{n\to\infty}[k_\Omega(z,z_n)-k_\Omega(z_0,z_n)]<\tfrac12\log R\right\},$$

where $z_n\to\partial\Omega$. Sequential-horosphere conjecture. The domain $\Omega$ has the Denjoy–Wolff property if and only if the closure of every sequential horosphere touches $\partial\Omega$ at exactly one point. This extends the corresponding equivalence known for simply connected planar domains to bounded convex domains in several complex variables.