Voisin's conjecture on general type varieties and approximate elliptic connectedness

Let $X$ be a projective variety embedded in projective space. It is approximately elliptically connected in the naïve sense if it can be connected by elliptic curves contained in arbitrarily small Euclidean neighborhoods of $X$ in the ambient projective space. Voisin's conjecture. A variety of general type is not approximately elliptically connected in the naïve sense. The statement is presented as a crucial condition for relating approximately elliptically connected varieties to Campana's special varieties, but no proof or disproof is supplied in the source.