  We consider a non-minimally coupled curvature-matter gravity model (NMC) with
a Weyl connection, a theory referred to as non-minimally coupled Weyl
connection gravity (NMCWCG). The Weyl connection is an affine connection that
is not compatible with the metric, and involves a vector field. Assuming a
vacuum expectation value for the vector field and a matter Lagrangian that only
contains the contributions of the vacuum energy, we show that the model admits
solutions in the space-form with a reference curvature that can be fine-tuned
to much smaller values than the contribution of the matter fields. This shows
that, at least in principle, the model admits a workable cosmological
description.
