The secant-variety regularity and projective-normality conjecture for embedded curves

Work over an algebraically closed field of characteristic $0$. Let $C\subset\mathbb{P}^n$ be a linearly normal embedding of a smooth curve of genus $g$ by a line bundle $L$, and let $\Sigma_k$ denote its $k$th secant variety. If

$$\operatorname{c}_1(L)\geq 2g+1+2k,\qquad k\geq0,$$

then the secant-variety regularity and projective-normality conjecture. The ideal sheaf $\mathcal I_{\Sigma_k}$ is $(2k+3)$-regular and $\Sigma_k$ is projectively normal.

This extends the established regularity result for the first secant variety and is suggested by the hypersurface degree of the secant variety of an elliptic normal curve. The assertion is presented as a conjecture, and no resolution is given in the source.