  The PL geometric category of a polyhedron $P$, denoted $\hbox{plgcat}(P)$,
provides a natural upper bound for the Lusternik--Schnirelmann category and it
is defined as the minimum number of PL collapsible subpolyhedra of $P$ that
cover $P$. In dimension 2 the PL geometric category is at most~3. It is easy to
characterize/recognize $2$-polyhedra $P$ with $\hbox{plgcat}(P) = 1$. Borghini
provided a partial characterization of $2$-polyhedra with $\hbox{plgcat}(P) =
2$. We complement his result by showing that it is NP-hard to decide whether
$\hbox{plgcat}(P)\leq 2$. Therefore, we should not expect much more than a
partial characterization, at least in algorithmic sense. Our reduction is based
on the observation that 2-dimensional polyhedra $P$ admitting a shellable
subdivision satisfy $\hbox{plgcat}(P) \leq 2$ and a (nontrivial) modification
of the reduction of Goaoc, Pat\'{a}k, Pat\'{a}kov\'{a}, Tancer and Wagner
showing that shellability of $2$-complexes is NP-hard.
