  A generic feature of scalar extensions of general relativity is the coupling
of the scalar degrees of freedom to the trace $T$ of the energy-momentum tensor
of matter fields. Interesting phenomenology arises when the trace becomes
positive---when pressure exceeds one third of the energy density---a condition
that may be satisfied in the core of neutron stars. In this work, we study how
the positiveness of the trace of the energy-momentum tensor correlates with
macroscopic properties of neutron stars. We first show that the compactness for
which $T=0$ at the stellar center is approximately equation-of-state
independent, and given by $C = 0.262_{-0.017}^{+0.011}$ (90% confidence
interval). Next, we exploit Bayesian inference to derive a probability
distribution function for the value of $T$ at the stellar center given a
putative measurement of the compactness of a neutron star. This investigation
is a necessary step in order to use present and future observations of neutron
star properties to constrain scalar-tensor theories based on effects that
depend on the sign of $T$.
