Eisenbud–Koh–Stillman secant-variety determinantal conjecture for embedded curves

Let $X$ be a smooth curve and let $L$ be a very ample line bundle defining an embedding of $X$. For each integer $k$, let $Sec^kX$ denote the $k$-th secant variety. Eisenbud–Koh–Stillman conjecture. There is a bound on the degree of $L$ such that $Sec^kX$ is ideal theoretically defined by the $(k+2)\times(k+2)$ minors of a matrix of linear forms. The conjecture predicts determinantal equations for secant varieties of sufficiently positive embedded curves; the cited result of M. S. Ravi gives a set-theoretic version under the bound $\deg L\geq 4g+2k+3$, while the ideal-theoretic assertion remains unresolved in the source.