The determinantal generation conjecture for secant varieties

Let $X \subset \mathbb{P}^r$ be a projective scheme embedded by the complete linear series of a sufficiently ample line bundle, and let $\operatorname{Sec}^k(X)$ denote the Zariski closure of the union of the linear spaces spanned by collections of $k+1$ points on $X$, where $k$ is a positive integer. Determinantal generation conjecture. The homogeneous ideal of $\operatorname{Sec}^k(X)$ is generated by the $(k+2)$-minors of a $1$-generic matrix of linear forms. This is proposed as a natural generalization of the theorem that sufficiently ample line bundles give determinantal presentations; its validity for arbitrary projective schemes and all positive integers $k$ remains open.