Progress on generalized star-height problem? The (generalized) star height of a language is the minimum nesting of Kleene stars required to represent the language by an extended regular expression. Recall that an extended regular expression over a finite alphabet $A$ satisfies the following:

(1) $\emptyset, 1$ and $a$ are extended regular expressions for all $a\in A$

(2) For all extended regular expressions $E,F$; 
 $E\cup F$, $EF$, $E^*$ and $E^c$ are extended regular expressions

One phrasing of the generalized star height problem is whether there is an algorithm to compute the minimum generalized star height.  With regards to this problem I have a few questions.


Has there been any recent progress (or research interest) concerning this problem? I know a number of years ago that Pin Straubing and Thérien published some papers in this area. 
The restricted star height problem was resolved in 1988 by Hashiguchi but the generalized version  (as far as I know) is still open. Does anyone have any intuition as to why this might be the case?


A link that might be helpful is the following:  starheight
Regarding your second question, an explanation why the generalized star height problem is less accessible than the star height problem is the following: Already Eggan's seminal paper in 1963 contained languages of (ordinary) star height $k$, for each $k\ge 0$. Only a few years later, McNaughton, and, independently, Déjean and Schützenberger, found examples over binary alphabets. This made clear what the problem "is about". During the years that followed, there was a more or less steady flow of published results in the area of the ordinary star height problem. This gave an ever increasing body of published examples, counterexamples and phenomena surrounding this problem.

In contrast, after some fifty years now, we don't know whether there is any regular language of star height at least two. So we do not even know whether there is a need for a decision procedure after all. This "complete lack of examples" indicates that it is extremely difficult to get a grip on this problem.