Decidability of dot-depth 2

The following problem is well-known and still open for around four decades in the TCS community.

Focusing only on the problem statement: a language L⊆Σ* is of dot-depth two if it is a Boolean combination of languages of the form Σ₁*a₁Σ₂*a₂...Σ_k*a_kΣ_k+1* for some Σ_i⊆Σ and a_i∈Σ. For example, (a+b)*a(b+c)*∩a*b(a+b)* is of dot-depth two.

The question: is the following problem decidable?

• Input: a finite automaton recognizing some language L.
• Output: is L of dot-depth two?

Note: for the case of piecewise testable languages, that is, the case with Σ_i=Σ for each i, there exist several polytime algorithms, the most recent being this one:

http://link.springer.com/chapter/10.1007%2F978-3-642-38771-5_26

It would be interesting to see whether this new proof can be changed to cover a broader class of languages.