Clemens's single-vanishing-cycle tube mapping conjecture

Let $X\subseteq \mathbb P^N$ be a smooth projective variety of dimension $n$, let $Y=X\cap H$ be a smooth hyperplane section, let $U\subseteq (\mathbb P^N)^*$ be the open set of smooth hyperplanes, and fix a base point $t\in U$. Let $\alpha_0\in H_{n-1}(Y,\mathbb Z)_{\textup{van}}$ be a primitive vanishing cycle. Clemens's conjecture. The map

$$\{([l],\alpha_0)\mid [l]\in \pi_1(U,t),\ l_*\alpha_0=\alpha_0\}\to H_n(X,\mathbb Z)_{\textup{prim}}$$

has a cofinite image. The conjecture restricts Schnell's tube mapping theorem to tubes over a single primitive vanishing cycle; the paper states that it proves the conjecture for cubic threefolds using Abel–Jacobi maps.