Han–Lee–Moon–Park conjecture on rank-three quadrics of sufficiently ample embeddings

Let $X$ be a projective scheme and let $L$ be a sufficiently ample line bundle on $X$. A quadric of rank $3$ means a degree-two equation whose associated quadratic form has rank $3$. Han–Lee–Moon–Park conjecture. The ideal $I(X,L)$ is generated by quadrics of rank $3$. The analogous statement for $I(X,L^{\otimes 2})$ is known, while the conjecture for $I(X,L)$ remains open according to the source.