Generic subtypical Segre identifiability conjecture

Let $d \ge 3$ and $n_1 \ge \cdots \ge n_d \ge 2$. Let $\mathcal{S}_{\mathbb{F}}=\operatorname{Seg}(\mathbb{P}\mathbb{F}^{n_1}\times\cdots\times\mathbb{P}\mathbb{F}^{n_d})$ be the Segre variety in $\mathbb{P}(\mathbb{F}^{n_1}\otimes\cdots\otimes\mathbb{F}^{n_d})$. A decomposition is generically $r$-identifiable when a generic tensor of rank $r$ has a unique decomposition. Generic subtypical Segre identifiability conjecture. The variety $\mathcal{S}_{\mathbb{F}}$ is generically $r$-identifiable for every strictly subtypical rank $r<\overline r_{\mathcal{S}_{\mathbb{F}}}$, except in the six listed cases: (1) $n_1>\prod_{k=2}^d n_k-\sum_{k=2}^d(n_k-1)$ and $r\ge\prod_{k=2}^d n_k-\sum_{k=2}^d(n_k-1)$; (2) $\operatorname{Seg}(\mathbb{P}\mathbb{F}^{4}\times\mathbb{P}\mathbb{F}^{4}\times\mathbb{P}\mathbb{F}^{3})$ with $r=5$; (3) $\operatorname{Seg}(\mathbb{P}\mathbb{F}^{n}\times\mathbb{P}\mathbb{F}^{n}\times\mathbb{P}\mathbb{F}^{2}\times\mathbb{P}\mathbb{F}^{2})$ with $r=2n-1$; (4) $\operatorname{Seg}(\mathbb{P}\mathbb{F}^{4}\times\mathbb{P}\mathbb{F}^{4}\times\mathbb{P}\mathbb{F}^{4})$ with $r=6$; (5) $\operatorname{Seg}(\mathbb{P}\mathbb{F}^{6}\times\mathbb{P}\mathbb{F}^{6}\times\mathbb{P}\mathbb{F}^{3})$ with $r=8$; or (6) $\operatorname{Seg}(\mathbb{P}\mathbb{F}^{2})^{\times 5}$ with $r=5$. The first three cases generically admit infinitely many decompositions; case (4) generically admits two complex decompositions, case (5) is expected to generically admit six complex decompositions, and case (6) generically admits two complex decompositions. The conjecture was initially stated over $\mathbb{C}$; computational results establish it there when $n_1\cdots n_d\le 15000$, with an extension to $\mathbb{R}$ described in the source.