Exponential growth of higher filling functions

Determine whether, when k is minimal such that a real triangulable Lie group G has the k-SOL obstruction, the higher filling function of degree k of G has exponential growth.

Background

The paper recalls that the SOL obstruction implies exponential growth of the ordinary Dehn function and asks whether the analogous higher-dimensional obstruction produces exponential growth in the corresponding degree-k filling function. This would extend the relationship developed in the paper between obstruction conditions, filling growth, and large homotopy or homology in asymptotic cones.