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 +0.011 −0.017 (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 .