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Bright basal reflectors in radargram from the Mars Advanced Radar for Subsurface and Ionosphere Sounding (MARSIS) of the Martian south polar layered deposits (SPLD) have been interpreted to be evidence of subglacial lakes. However, this interpretation is difficult to reconcile with the low Martian geothermal heat flow and the frigid surface temperature at the south pole. We conduct a comprehensive thermophysical evolution modeling of the SPLD and show that subglacial lakes may only form under exceptional circumstances. Subglacial lakes may form if the SPLD contains more than 60 % dust volumetrically or extremely porous ice (>30 %), which is unlikely. A thick (>100 m) layer of dirty ice (>90% dust) at the base of the SPLD may also enable basal melting, resembling a sludge instead of a lake. Other scenarios enabling subglacial lakes in the SPLD are equally unlikely, such as recent magmatic intrusions at shallow depths., Submitted to GRL