New decidable upper bound of the second level in the Straubing-Thérien concatenation hierarchy of star-free languages

In a recent paper we gave a counterexample to a longstanding conjecture concerning the characterization of regular languages of level 2 in the Straubing-Thérien concatenation hierarchy of star-free languages. In that paper a new upper bound for the corresponding pseudovariety of monoids was implicitly given. In this paper we show that it is decidable whether a given monoid belongs to the new upper bound. We also prove that this new upper bound is incomparable with the previous upper bound.