  In [3], after defining notions of LS category in the simplicial context, the
authors show that the geometric simplicial LS category is non-decreasing under
strong collapses. However, they do not give examples where it increases
strictly, but they conjecture that such an example should exist, and thus that
the geometric simplicial LS category is not strong homotopy invariant. The
purpose of this note is to provide with such an example. We construct a
simplicial complex whose simplicial and geometric simplicial LS categories are
different, and using this, we provide an example of a strong collapse that
increases the geometric simplicial LS category, thus settling the geometric
simplicial LS category not being strong homotopy invariant.
