  An anti-correlation between the central density of the dark matter halo
($\rho_{150,\ {\rm DM}}$) and the pericentric distances ($r_{p}$) of the Milky
Way's (MW's) dwarf spheroidal galaxies (dSphs) has been reported in the
literature. The existence and origin of such anti-correlation is however
controversial, one possibility being that only the densest dSphs can survive
the tidal field towards the centre of our Galaxy. In this work, we place
particular emphasis on quantifying the statistical significance of such
anti-correlation, by using available literature data in order to explore its
robustness under different assumptions on the MW gravitational potential, and
for various derivations of $\rho_{150}$ and $r_{p}$. We consider models in
which the MW is isolated and has a low ($8.8\times10^{11}\,M_{\odot}$) and high
($1.6\times10^{12}\, M_{\odot}$) halo mass, respectively, as well as
configurations in which the MW's potential is perturbed by a Large Magellanic
Cloud (LMC) infall. We find that, while data generally support models in which
the dSphs' central DM density decreases as a function of their pericentric
radius, this anti-correlation is statistically significant at $3\sigma$ level
only in $\sim$12$\%$ of the combinations of $\rho_{150}$ and $r_{p}$ explored.
Moreover, including the impact of the LMC's infall onto the MW weakens or even
washes away this anti-correlation, with respect to models in which the MW is
isolated. Our results suggest that the strength and existence of such
anti-correlation is still debatable: exploring it with high-resolution
simulations including baryonic physics and different DM flavours will help us
to understand its emergence.
