In order to prove some Liouville pair is a Weinstein pair, we need to know that the stop is good.
If our space is like $\R \times \R_-$, one factor of space is cutting off some factor, but leaving some other factors intact. How does the fiber look like? It would be just the stop-fiber from the relevant factor, times the entire space from the non-partipating factor. So to show the Weinstein-ness, one just need to show that the two factors are. Now, where is the participating factor? It is about some smoothable function, that only involves center of mass variables. So, it is as if in the cotangent bundle case.
Now, how about the other factors? What do we need to show? What do we have already? Those other factor is locally a product, which we assume is Weinstein already.