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We propose and test an exchange gas technique for improving the cooldown times of cryocooled gravitational-wave interferometers. The technique works by utilizing low-pressure dry nitrogen gas to create a path for heat conduction to test masses while protecting the rest of the in-vacuum equipment from unwanted heat leakage. We show that the technique is capable of shortening the total wait time to reach the operating temperature by a factor of 3.5. Additionally, our tests show that the improvement in the heat transfer rate can be predicted to be within 10% error by using the Sherman-Lees interpolation equation. The technique is compatible with vibration isolation requirements of the cryogenic shielding of 124 K silicon interferometers and has the potential to improve the iteration time for research and development. The scalability of the prototype, the ability to predict the heat conduction, and the simplicity of the engineering make the strategy a good candidate to be included in the cryogenic design of future cryocooled gravitational-wave interferometers. The findings mark a first step in the investigation for a strategy to mitigate ice formation on the interferometer optics during initial cooldown.