Chiantini–Ottaviani–Vannieuwenhoven conjecture on generic identifiability of Segre varieties

Let $\mathcal{S}_\mathbb{C}=\operatorname{Seg}(\mathbb{C}^{n_1}\times\mathbb{C}^{n_2}\times\cdots\times\mathbb{C}^{n_d})\subset\mathbb{C}^{\Pi}$ be a Segre variety with $d\geq 3$ and $n_1\geq n_2\geq\cdots\geq n_d\geq 2$. A Segre variety is generically $r$-identifiable if, outside a proper Zariski-closed exceptional set containing the lower-rank locus, every tensor in its $r$-th secant variety has a unique rank-$r$ decomposition.

Chiantini–Ottaviani–Vannieuwenhoven conjecture. $\mathcal{S}_\mathbb{C}$ is generically $r$-identifiable if

$$r<\frac{\Pi}{\Sigma+1},$$

unless $(n_1,n_2,\ldots,n_d)$ is one of $(4,4,3)$, $(4,4,4)$, $(6,6,3)$, $(n,n,2,2)$ with $n\in\mathbb{N}$, $(2,2,2,2,2)$, or satisfies

$$n_1>\prod_{k=2}^d n_k-\sum_{k=2}^d(n_k-1).$$

This conjecture aims to give a nearly complete picture of generic tensor identifiability over $\mathbb{C}$, which is important for uniqueness and interpretation of tensor rank decompositions. The supplied text presents it as a conjecture but gives no explicit resolution status.