Let $S$ be a finite semigroup. A factorization tree of a product $w_1\cdots w_n$ in $S$ is a finite ordered tree labeled by elements of $S$, such that each leaf is labeled by the $w_i$ in their order in the product, and such that each internal node is labeled by the product of their leaves (in their order). A factorization tree is Ramseyan if each internal node of arity $3$ or more has all its children labeled by the same idempotent.
Theorem. (Simon) Each product in a finite semigroup $S$ has a Ramseyan factorization tree of height at most $9|S|$.
The bound was later improved to at least $3|S|$. I found in the two papers [1] and [2] the mention that this bound can be improved to $3|S|-1$. Here is a quote from [2]:
The improved bound $3 |S| − 1$ can be obtained from the current proof by some additional rotation technique as in the proof of Lemma 2. We postpone this proof to a forthcoming journal version of this article.
However, I found no such journal version and was unable to reconstruct the proof for the bound $3|S|-1$. We can make a recurrence from the lowest Green $\mathcal{J}$-classes to the highest instead of the other way around but I still obtained the bound $3|S|$ instead of $3|S|-1$.
Can someone give a proof of this result or a reference?
[1] Factorization forests for infinite words and applications to countable scattered linear orderings (Colcombet, 2010)
[2] The Height of Factorization Forests (Kufleitner, 2008)
The published version of [2] is
M. Kufleitner, The height of factorization forests. In E. Ochmanski and J. Tyszkiewicz, editors, Proc. 33rd Symp., Mathematical Foundations of Computer Science 2008, volume 5162 of Lecture Notes in Comput. Sci., pages 443–454. Springer, 2008.
It contains a detailed proof of the result you are looking for.