Degree bounds for equations and syzygies of secant varieties of curves

Let $X$ be a smooth curve embedded by a line bundle $L$, and let $Sec^{k-1}X$ denote its $(k-1)$-st secant variety. Secant-variety degree-bound conjecture. If $\deg(L)\geq 2g+2k$, then $Sec^{k-1}X$ is defined as a scheme by forms of degree $k+1$. If $\deg(L)\geq 2g+2k+1$, then $Sec^{k-1}X$ satisfies condition $(K_{k+1})$. These bounds combine degree estimates from constructions cited in the source; the supplied text presents the assertion as a conjectural statement, with no resolution evidence provided.