The preformed-pairs theory of pseudogap physics in high-$T_C$ superconductors predicts a nonanalytic $T$-dependence for the $ab$-plane superfluid fraction, $rho_S$, at low temperatures in underdoped cuprates. We report high-precision measurements of $rho_S(T)$ on severely underdoped YBa$_2$Cu$_3$O$_{6+x}$ and Y$_{0.8}$Ca$_{0.2}$Ba$_2$Cu$_3$O$_{6+x}$ films. At low $T$, $rho_S$ looks more like $1 - T^2$ than $1 - T^{3/2}$, in disagreement with theory.
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