The generic tensor-rank conjecture for multipartite quantum states

Let $\mathcal{H}=\mathbb{C}^{d_1}\otimes\cdots\otimes\mathbb{C}^{d_n}$ be the Hilbert space of multipartite quantum states. The expected tensor rank is the smallest $k$ for which the $k$-secant variety fills the ambient projective space; equivalently, it is the ceiling of the ambient dimension divided by the dimension of a general rank-one parameter count. Generic tensor-rank conjecture. For a general multipartite quantum state, the generic tensor rank is

$$\operatorname{rk}_{\text{gen}}=\left\lceil\frac{\prod_{i=1}^n d_i}{\sum_{i=1}^n(d_i-1)+1}\right\rceil,$$

except for $\mathbb{C}^{4\times4\times3}$, $\mathbb{C}^{(2i+1)\times(2i+1)\times3}$, and $\mathbb{C}^{(i+2)\times(i+2)\times2\times2}$ with $i\in\mathbb{Z}^+$, where it is the expected rank plus one. This is the expected-dimension prediction for secant varieties of Segre varieties. The supplied text records the exceptional cases but does not state whether the conjecture has been resolved.