I also Had a quick look (maybe a little less quick), and although I very much like the other answer, which illustrates that it may be difficult to fix, I may have found a more specific error, which may be more helpfull as an answer to the question.

Firstly, I am a little confused as to what constitutes a stratification. I see two possibilities:

1. The one which is actually defined which allows the following stratification: $S_1=S_2=D^2\times [0,1]$ and they are glued along a closed disc in the interior of $D^2\times \{1\}$ of $S_1$ and the same disc in the interior of $D^2 \times \{0\}$ in $S_2$.

2. The one which I think is implied at some points: $S_i$ and $S_{i+1}$ may only be identified such that $U(S_i) \cup L(S_{i+1})$ is in fact a sub-surface in the 3-manifold.

I will describe my problems related to both definitions:

In the proof of prop 5.8 parts (2-3-4) he attaches "3-cell"s (I would write 3-disc as to avoid confusion with CW complex attachments of cells, or attach both a 2-cell and a 3-cell) $W$, and extends the stratification.

If we work under definition 2) above then this seems generally impossible because you would often also have to attach it at the top of $S_{i+1}$ to get the extra surface assumption in 2).

If we work under definition 1) above then this doesn't even make the new $F_{i+1}$ a surface in the simple example described above.