Perfect GO-spaces which have a perfect linearly ordered extension

It is an old problem posed by Bennett and Lutzer whether every perfect GO-space has a perfect orderable extension. As an approach to this problem, we prove that, for a perfect GO-space X, Xhas a perfect linearly ordered extension if and only if there is a σ-discrete subset F such that GOX(∅, X − F, F, ∅) is perfect, where GOX(∅, X − F, F, ∅) is the ordered set X with the topology defined so that every point in F is isolated and every point in X − F has the usual interval neighborhood base.