Hausdorff gaps reconstructed from Luzin gaps

We consider a question: Can a given AD-family be ADR for two orthogonal uncountable towers? If $b>{\omega }_1$, then we rebuilt any AD-family of the cardinality ${\omega }_1$ onto a Hausdorff pre-gap. Moreover, if a such AD-family is a Luzin gap, then we obtain a Hausdorff gap. Under $b={\omega }_1$, a similar rebuilding is impossible.