Determinantal presentation conjecture for secant varieties of factored line bundles

Let $C$ be a curve of genus $g$, and let $L$ be a line bundle that factors as

$$L=L_1\otimes L_2.$$

For an integer $k$, write $\operatorname{Sec}^k(C)$ for the $k$-th secant variety of the image of $C$ under the linear series associated to $L$. A variety is determinantally presented if its ideal is generated by minors of a matrix of linear forms. Determinantal presentation conjecture. There is a constant $k_0$, depending on the genus of $C$ and the degrees of the $L_i$, such that $\operatorname{Sec}^k(C)$ is determinantally presented for $k\leq k_0$. This conjecture predicts determinantal equations for an initial range of secant varieties whenever the embedding line bundle factors as a tensor product. The supplied text gives no resolution or evidence establishing a value of $k_0$, so the conjecture is recorded as open.