  We study how rapid rotation influences the relation between the normalized
moment of inertia $\bar{I}$ and quadrupole moment $\bar{Q}$ for scalarized
neutron stars. The questions one has to answer are whether the EOS universality
is preserved in this regime and what are the deviations from general
relativity. Our results show that the $\bar{I}-\bar{Q}$ relation is nearly EOS
independent for scalarized rapidly rotating stars, but the differences with
pure Einstein's theory increase compared to the slowly rotating case. In
general, smaller negative values of the scalar field coupling parameters
$\beta$ lead to larger deviations, but these deviations are below the expected
accuracy of the future astrophysical observations if one considers values of
$\beta$ in agreement with the current observational constraint. An important
remark is that although the normalized $\bar{I}-\bar{Q}$ relation is quite
similar for scalar-tensor theories and general relativity, the unnormalized
moment of inertia and quadrupole moment can be very different in the two
theories. This demonstrates that although the $\bar{I}-\bar{Q}$ relations are
potentially very useful for some purposes, they might not serve us well when
trying to distinguish between different theories of gravity.
