Markus conjecture on completeness and parallel volume

Let $B$ be a connected affine manifold, with affine structure group $\mathrm{GL}_n(\mathbb{Z})\ltimes H^n$, and let completeness mean that all geodesics on $B$ are parametrised by the entire real line. A volume is parallel if it is preserved by the affine holonomy. Markus conjecture. An affine manifold is complete if and only if it has parallel volume. The conjecture concerns the relationship between the global geodesic structure of an affine manifold and its affine holonomy; the source does not state a resolution status.