Conjecture on defective secant varieties of Segre-Veronese varieties embedded by $\mathcal{O}(1,2)$

Let $X_{m,n}$ be the Segre-Veronese variety ${\mathbb P}^{m,n}$ embedded by the morphism given by $\mathcal{O}(1,2)$. For a positive integer $s$, let $\sigma_s(X_{m,n})$ denote the $s$-th secant variety of $X_{m,n}$. The pair $(m,n;1,2)$ is called unbalanced when the unbalancedness condition used in case (a) holds.

The defective secant-variety conjecture. The variety $\sigma_s(X_{m,n})$ is defective if and only if $(m,n,s)$ falls into one of the following cases:

• $(m,n;1,2)$ is unbalanced and

$${n+2 \choose 2}-n < s < \min\left\{m+1,{n+2 \choose 2}\right\};$$

• $(m,n,s)=(2,2k+1,3k+2)$ with $k\geq 1$;
• $(m,n,s)=(4,3,6)$.

The conjecture proposes a complete classification of defective secant varieties for this Segre-Veronese embedding; the paper's experiments and the explicitly established defectivity of $\sigma_6(X_{4,3})$ motivate it, while no resolution is supplied in the given source.