Asymptotic equations and syzygies conjecture for secant varieties

Let $X$ be a smooth variety, let $L$ be an ample line bundle on $X$, and fix an integer $k\geq 1$. For sufficiently large $n$, the line bundle $L^n$ defines an embedding of $X$; write $Sec^kX$ for the resulting $k$-th secant variety. Asymptotic secant-variety conjecture. For all $n\gg0$, $Sec^kX$ is ideal theoretically defined by forms of degree $k+2$, and furthermore satisfies condition $(K_{k+2})$. This conjecture proposes uniform asymptotic control of both equations and syzygies of secant varieties; the supplied source gives no resolution status beyond presenting it as a basic conjectural statement.