  We consider one of the weakest variants of cost register automata over a
tropical semiring, namely copyless cost register automata over $\mathbb{N}$
with updates using $\min$ and increments. We show that this model can simulate,
in some sense, the runs of counter machines with zero-tests. We deduce that a
number of problems pertaining to that model are undecidable, in particular
equivalence, disproving a conjecture of Alur et al. from 2012. To emphasize how
weak these machines are, we also show that they can be expressed as a
restricted form of linearly-ambiguous weighted automata.
