Laumon's purity conjecture for compactified Jacobians

Let $C$ be a projective geometrically integral algebraic curve over $k$ whose singularities are all planar, and let $\overline{\operatorname{Jac}}_{C}$ be its compactified Jacobian. Laumon's purity conjecture. The normalization of $\overline{\operatorname{Jac}}_{C}$ is cohomologically pure. This is presented as a reformulation related to the purity conjectures for affine Springer fibers and compactified Jacobians, and it motivates a possible deformation approach; the general assertion remains open in the source.