Border rank two does not force a tensor-square rank saving

An explicit smooth, irreducible, nondegenerate curve $X \subset \mathbb{P}^{11}$
and a point $p$ satisfy $\underline{R}_X(p) = 2$, $R_X(p) = 3$, and
$R_{X \times X}(p \otimes p) = 9$. Thus the implication conjectured by Ballico, Bernardi,
Gesmundo, Oneto and Ventura is false for its stated class of projective
varieties. More generally,
for every $r \ge 3$ and every $m \ge 2$, a smooth projected rational normal
curve has a point of border rank two and rank $r$ whose first $m$ tensor powers
have ranks $r, r^2, \ldots, r^m$. The proof is elementary: a projection
preserves independence of small sets of distinct curve points while
identifying a high-rank tangent vector with a sum of $r$ curve points. All
tensor products retain their separate factors.