  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.
