Gallego–Purnaprajna conjecture on very ampleness of twice a polarization

Let $X$ be a smooth Calabi-Yau threefold and $L$ a globally generated, ample line bundle. For a surface $S\in |L|$, write $|L\mathbin{\vert}_S|$ for the restriction of the linear system to $S$; a curve $C\in |L\mathbin{\vert}_S|$ is nonhyperelliptic if it is smooth and nonhyperelliptic. Gallego–Purnaprajna conjecture. The line bundle $2L$ is very ample and the morphism it defines embeds $X$ as a projectively normal variety if and only if there is a smooth nonhyperelliptic curve $C\in |L\mathbin{\vert}_S|$ for some $S\in |L|$. This conjecture proposes a criterion for very ampleness and projective normality on Calabi-Yau threefolds; the supplied source does not indicate whether it has been resolved.