Uniqueness conjecture for Riemannian solutions of the metric-affine Bach problem

In the Bach case, the weights satisfy

$$c_3=1\quad\operatorname{and}\quad c_j=0,\quad\forall j\ne3.$$

The Bach action is the squared norm of the Weyl curvature, and the problem is varied independently with respect to the metric and connection. A Riemannian solution is a solution with metric-compatible connection, while a conformally flat space has zero Weyl curvature and an Einstein space has Ricci curvature proportional to the metric. Bach-case uniqueness conjecture. The only Riemannian solutions of the metric-affine Bach problem are conformally flat spaces and Einstein spaces. This case is exceptional because every spacetime with zero Weyl curvature is automatically a solution, while the remaining Riemannian equations are still heavily overdetermined.