Border Comon's conjecture for minimal border rank

Let $F\in (\mathbb{C}^n)^{\otimes d}$ be a concise symmetric tensor of minimal border tensor rank, and let $p_F$ be its corresponding polynomial. Here ${\underline{{\bf rk}}}(F)$ denotes the border tensor rank of $F$, and ${\underline{{\bf rk}}}_S(p_F)$ denotes the symmetric border rank of $p_F$.

Border Comon's conjecture for minimal border rank. If

$${\underline{{\bf rk}}}(F)=n,$$

then

$${\underline{{\bf rk}}}_S(p_F)={\underline{{\bf rk}}}(F).$$

This is the minimal-border-rank case of border Comon's conjecture, which asks whether border rank and symmetric border rank agree for symmetric tensors. The problem is largely open, although the paper establishes it for large classes of concise tensors, including all tame tensors and all tensors when $n\leq d+1$.