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The fundamental limits of the microwave noise performance of high electron-mobility transistors (HEMTs) are of scientific and practical interest for applications in radio astronomy and quantum computing. Self-heating at cryogenic temperatures has been reported to be a limiting mechanism for the noise, but cryogenic cooling strategies to mitigate it, for instance, using liquid cryogens, have not been evaluated. Here, we report microwave noise measurements of a packaged two-stage amplifier with GaAs metamorphic HEMTs immersed in normal and superfluid [Formula: see text]He baths and in vacuum from 1.6 to 80 K. We find that these liquid cryogens are unable to mitigate the thermal noise associated with self-heating. Considering this finding, we examine the implications for the lower bounds of cryogenic noise performance in HEMTs. Our analysis supports the general design principle for cryogenic HEMTs of maximizing gain at the lowest possible power.