To the best of the author's knowledge, the problem is still open.
However, to our knowledge this is not fully resolved.
Indeed, to our knowledge, cardinality restrictions on Berg spaces have received no prior attention outside the metric context.
Give a proof of Theorem 2 which requires no knowledge of the boundary values of $f$.
The knowledge of the invariant subspaces of an operator helps us to visualize its action.
We presume a basic knowledge of large cardinals and forcing.