Whether self-organized criticality needs hidden fine tuning

Determine whether physically realizable driven dissipative systems exist whose steady state is critical (avalanche distributions with a cutoff diverging with system size) for an open set of control parameters, as opposed to requiring the double limit of drive rate $h \to 0$ and bulk dissipation $\epsilon \to 0$ that maps self-organized criticality onto a tuned absorbing-state transition. Answer: yes with a model or experiment, or no with a general argument.

A model or experiment showing size-divergent avalanche cutoffs over a finite range of drive and dissipation rates, or a proof that the double limit is necessary for a stated class.