Bucher–Connell–Lafont simplicial-volume conjecture for affine manifolds

Let $M$ be a closed manifold supporting an affine structure, meaning a closed manifold admitting an atlas whose transition maps are restrictions of affine isomorphisms. Let $\|M\|$ denote the simplicial volume of $M$. Bucher–Connell–Lafont conjecture. The simplicial volume vanishes:

$$\|M\|=0.$$

This conjecture was formulated by Bucher, Connell and Lafont and is presented as open. It would give a broad geometric strengthening related to the Chern conjecture, whose Euler-characteristic conclusion is known in several special cases.