The violation of the weak energy condition, Is it generic of spontaneous scalarization ?

It was recently shown by Whinnett & Torres \cite{WT} that the phenomenon of spontaneous scalarization (SC) in compact objects (polytropes) was accompanied also by a {\it spontaneous violation of the weak energy condition} (WEC). Notably, by the encounter of negative-energy densities at several star places as measured by a static observer. Here we argue that the negativeness of such densities is not generic of scalar tensor theories of gravity (STT). We support this conclusion by numerical results within a class of STT and by using three realistic models of dense matter. However, we show that the ``angular parts'' of the additional conditions for the WEC to hold $\rho_{\rm eff} + T^{i ({\rm eff})}_{i} \geq 0$ tend to be ``slightly violated'' at the outskirts of the star.