Perfect-rank non-identifiability conjecture for Segre varieties

Let $d\ge3$ and $n_1\ge n_2\ge\cdots\ge n_d\ge2$. Let $\mathcal S=\operatorname{Seg}(\mathbb P\mathbb C^{n_1}\times\cdots\times\mathbb P\mathbb C^{n_d})$ be the Segre variety in $\mathbb P(\mathbb C^{n_1}\otimes\cdots\otimes\mathbb C^{n_d})$, and suppose that $\overline r_{\mathcal S}$ is an integer. Perfect-rank non-identifiability conjecture. The variety $\mathcal S$ is not generically $\overline r_{\mathcal S}$-identifiable, except when it is $\operatorname{Seg}(\mathbb P\mathbb C^5\times\mathbb P\mathbb C^4\times\mathbb P\mathbb C^3)$ or $\operatorname{Seg}(\mathbb P\mathbb C^3\times\mathbb P\mathbb C^2\times\mathbb P\mathbb C^2\times\mathbb P\mathbb C^2)$. This concerns the perfect case, where the expected generic rank is integral, and complements the subtypical identifiability results discussed in the source.