Finite Ramsey degrees of ages of infinite-dimensional classical spaces

Determine whether the ages of countably infinite-dimensional symplectic, orthogonal, or unitary spaces over finite fields have finite Ramsey degree for every member of the age.

Background

The paper proves that the age of a countably infinite-dimensional symplectic space over GF(2) does not have the Ramsey property. The authors then ask whether the weaker property of finite Ramsey degrees nevertheless holds for the ages of infinite-dimensional symplectic, orthogonal, and unitary spaces. A positive answer would be relevant to the metrizability of universal minimal flows of the corresponding automorphism groups.