Closedness of the subgroup H in the Boolean linear group construction

Determine whether, in the construction obtained by applying the argument of Theorem 3 to the free Boolean linear topological groups Blin(C1) and Blin(S2) (where C1 and S2 are zero-dimensional spaces derived from the Cantor set), the subgroup H with positive Katětov covering dimension dim0 H > 0 can be chosen to be closed in the ambient strongly zero-dimensional Boolean group G with linear topology.

Background

Section 4 establishes that covering dimension is not preserved by subgroups by constructing a strongly zero-dimensional Boolean group G containing a closed subgroup H with positive covering dimension, using the free Boolean topological groups B(C1) and B(S2) built over special zero-dimensional spaces C1 and S2 derived from the Cantor set. The closure of H is secured via Raikov completeness, which is preserved under products.

Remark 4 seeks an analogous example within the class of groups with linear topology by replacing B(C1) and B(S2) with the free Boolean linear topological groups Blin(C1) and Blin(S2). While this yields a strongly zero-dimensional Boolean group G with linear topology that contains a subgroup H with dim0 H > 0, the authors note that it is unclear whether H can be made closed because free Boolean linear groups over Dieudonné complete (even compact) spaces may fail to be complete, citing an explicit non-complete example. This leaves the closedness of H in such a construction unresolved.