  We investigate three models for the charge symmetry breaking (CSB) of the
nucleon-nucleon ($NN$) interaction (based upon $\rho$-$\omega$ mixing, nucleon
mass splitting, and phenomenology) that all reproduce the empirical value for
the CSB of the $^1S_0$ scattering length ($\Delta a_{CSB}$) accurately. We
reveal that these models make very different predictions for CSB in $^3P_J$
waves and examine the impact of this on some observable quantities of $A\geq 3$
nuclear systems. It turns out that the $^3$H-$^3$He binding energy difference
is essentially ruled by $\Delta a_{CSB}$ and not very sensitive to CSB from $P$
waves. However, the Coulomb displacement energies (which are the subject of the
Nolen-Schiffer anomaly) receive about 50% of their CSB contribution from $NN$
partial waves beyond $^1S_0$. Consequently, the predictions by the various CSB
models differ here substantially (10-20%). Unfortunately, the evaluation of the
leading Coulomb contributions carry a large uncertainty such that no
discrimination between the competing CSB models can presently be made. To
decide the issue we suggest to look into nuclear few-body reactions that are
sensitive to CSB of the nuclear force.
