  In our recent works we derived a chiral $O(q^4)$ two-pion exchange
nucleon-nucleon potential (TPEP) formulated in a relativistic baryon (RB)
framework, expressed in terms of the so called low energy constants (LECs) and
functions representing covariant loop integrations. We showed that the
expansion of these functions in powers of the inverse of the nucleon mass
reproduces most of the terms of the TPEP derived from the heavy baryon (HB)
formalism, but such an expansion is ill defined and does not converge at large
distances. In the present work we perform a study of the phase shifts in
nucleon-nucleon ($NN$) scattering for peripheric waves ($L\geq 3$), which are
sensitive to the tail of the potential. We assess quantitatively the
differences between the RB and HB results, as well as variations due to
different values of the LECs. By demanding consistency between the LECs used in
$\pi N$ and $NN$ scattering we show how $NN$ peripheral phase shifts could
constrain these values. We demonstrate that this idea, first proposed by the
Nijmegen group, favors a smaller value for the LEC $c_3$ than the existing
ones, when considering the TPEP up to order $q^4$ in the chiral expansion.
