Ramsey property for finite subspaces whose projection contains a distinguished subspace

Determine whether, for a countably infinite-dimensional vector space over GF(2), the family of finite subspaces U whose projection onto a fixed finite-dimensional subspace A1 contains A1 has the Ramsey property.

Background

After proving a positive Ramsey result for finite subspaces containing a fixed distinguished subspace, the paper considers the dual formulation involving a projection T from the ambient vector space onto A1. The family under consideration consists of finite subspaces whose image under T contains A1. The authors prove the result when the dimension of A1 is at most one and reduce the general case to a Ramsey property for n-space-tuples.