We experimentally test a theoretically promising amendment to the ratchet-up mechanism of the Paris Agreement. The ratchet-up mechanism prescribes that parties' commitments to the global response to climate change cannot decrease over time and our results confirm that its effect is detrimental. We design a public goods game to study whether an amendment to the mechanism that stipulates all agents to contribute at least a collective minimum to the public good which is based on the principle of the lowest common denominator promotes cooperation. We find that binding collective minimum contributions improve the effectiveness of the ratchet-up mechanism. Non-binding minimum contributions, in contrast, do not foster cooperation. Our data reveal conditional cooperative dynamics to explain the difference. If other participants contribute less than the collective minimum contributions, even initially cooperative participants start to negatively reciprocate such a form of non-compliance by contributing less.