  At the supranuclear densities achieved inside a neutron star, matter may
exhibit extreme properties. In particular, it may be the case that a suitable
average of the speed of sound squared exceeds the so-called conformal limit,
i.e., $\langle c_s^2 \rangle > 1/3$, a condition that is equivalent to the
positiveness of the trace of the energy-momentum tensor at the stellar center.
This property, that holds for highly compact neutron stars obeying many (but
not any) realistic equations of state, would turn these objects into
interesting laboratories for tests of several scalar extensions of general
relativity. In this paper, we investigate how rapid rotation influences the
superconformality of the averaged speed of sound squared and modified gravity
effects that depend thereupon, paying particular attention to scalar-tensor
theories prone to the spontaneous scalarization effect.
