Construct rational lasso expressions that yield syntactic or recurring FDFAs via the Brzozowski construction

Determine whether, for any given rational ω-expression over a finite alphabet, one can construct a rational lasso expression such that the Brzozowski construction for lasso automata applied to that expression yields an equivalent acceptor whose associated Family of DFAs (FDFA), in the sense of Angluin and Fisman, is syntactic or recurring rather than periodic.

Background

Angluin and Fisman introduced periodic, syntactic, and recurring FDFAs for ω-regular languages, showing that syntactic and recurring FDFAs can be exponentially smaller than periodic FDFAs. This paper provides a Brzozowski-style construction turning rational lasso expressions into finite lasso automata and develops Kleene theorems for lasso and ω-languages. The question seeks a refinement of the construction to systematically obtain lasso automata (and hence equivalent FDFAs) that are syntactic or recurring from rational ω-expressions, potentially improving automata size and efficiency.