  The tidal deformability probability distribution extracted from GW170817
alone, or including multi-messenger information, is confronted to astrophysical
and nuclear physics additional constraints within a semi-agnostic approach for
the dense matter equation of state. We use Bayesian statistics to combine
together low density nuclear physics data, such as the ab-initio predictions
based on $\chi$EFT interactions or the isoscalar giant monopole resonance, and
astrophysical constraints from neutron stars, such as the maximum mass of
neutron stars or the probability density function of the tidal deformability
$\tilde{\Lambda}$ obtained from the GW170817 event. The posteriors probability
distribution functions are marginalized over several nuclear empirical
parameters ($L_\textrm{sym}$, $K_\textrm{sym}$, $Q_\textrm{sat}$ and
$Q_\textrm{sym}$), as well as over observational quantities such as the
$1.4M_\odot$ radius $R_{1.4}$ and the pressure at twice the saturation density
$P(2n_\textrm{sat})$. The correlations between $L_\textrm{sym}$ and
$K_\textrm{sym}$ and between $K_\textrm{sat}$ and $Q_\textrm{sat}$ are also
further analyzed. Tension is found between the posteriors: the first one is
localized in the tidal deformability probability distribution itself, depending
whether multi-messenger analysis is included or not, and the second one is
between the observational data and the nuclear physics inputs. These tensions
impact the predictions for $L_\textrm{sym}$, $K_\textrm{sym}$ and $R_{1.4}$
with centroids which differ by 2-3$\sigma$. Implications for the nuclear
equation of state are also discussed.
