Generic uniqueness of best rank-one approximations for real tensors

Let $d\ge 3$ and let $n_j\ge 2$ for $j\in[d]$. A rank-one approximation of a tensor $\mathcal{T}\in\mathbb{R}^{n_1\times\cdots\times n_d}$ is a rank-one tensor minimizing its distance from $\mathcal{T}$ in the Frobenius norm. Write $\operatorname{Sym}(n,d)$ for the space of symmetric order-$d$ tensors in $(\mathbb{R}^n)^{\otimes d}$.

Generic uniqueness conjecture. Every $\mathcal{T}\in\mathbb{R}^{n_1\times\cdots\times n_d}$ has a unique rank-one approximation, unless $\mathcal{T}$ lies on a subvariety; moreover, every $\mathcal{T}\in\operatorname{Sym}(n,d)$ has a unique rank-one approximation, and this approximation is symmetric, unless $\mathcal{T}$ lies on a subvariety.

The conjecture predicts uniqueness outside an exceptional algebraic subset, together with symmetry of the unique approximation for symmetric tensors. The statement is presented as an expectation in the source, and no resolution is supplied here.