Hartshorne's deformation conjecture for subvarieties with ample normal bundle

Let $X$ be a variety and let $Y\subset X$ be a smooth subvariety. Its normal bundle $N_Y$ is a vector bundle, and it is ample when $\mathscr O(1)$ is ample on $\mathbb P(N_Y)$. Hartshorne's deformation conjecture. Some multiple of $Y$ deforms as a cycle in a family covering $X$. This is one of Hartshorne's conjectures on subvarieties with ample normal bundle; the supplied source gives no resolution status for this question.