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Most of the heat from radiators comes from convection (radiation would require them to glow). As such, covers that restrict airflow will reduce efficiency: you need to heat the cover up until its temperature matches that of the radiator, but that is challenging because you're transferring the heat via air rather than water.
So my bet would be yes. We had a (much uglier) cover on one of our roads and taking it off made a bit positive difference for the temperature the room gets to.
Does the hive think that this rad cover would prevent the rad from working anywhere close to efficiently ?