Expected-rank conjecture for finite tensor decompositions

Let $k\ge 3$ and let $n_i$ denote the dimensions of the factors, for $1\le i\le k$. Write $\mathbf{n}=(n_1,\ldots,n_k)$ and define

$$r_E(k,\mathbf{n})=\left\lceil\frac{\prod_{i=1}^k n_1}{1-k+\sum_{i=1}^k n_i}\right\rceil.$$

A generic tensor of order $k$ and rank $r$ is a tensor outside a proper exceptional subset. Expected-rank conjecture. A generic tensor of order $k\ge3$ and rank $r$ admits a finite number of decompositions into a sum of rank-one terms if $r<r_E(k,\mathbf{n})$. This is presented as the generally admitted extension of a symmetric-tensor result to unconstrained tensors; the source gives no resolution and mentions only partial results in the literature.