Clemens's cofinite-image conjecture for primitive vanishing cycles

Let $X$ be a smooth hypersurface and suppose that $n=\dim(X)$ is odd. Let $T_v$ be the connected component of the analytic covering space of the smooth hyperplane-section parameter space that consists of primitive vanishing cycles, and let the tube mapping send such cycles to primitive middle-dimensional homology classes of $X$. Clemens's conjecture. The restriction of the tube mapping to primitive vanishing cycles has a cofinite image. This asks whether the distinguished component of primitive vanishing cycles realizes the cofinite image known to occur for at least one connected component; it concerns the relationship between vanishing-cycle monodromy and the primitive cohomology of hypersurfaces.