  The observed anti-correlation between the central dark matter (DM) densities
of the bright Milky Way (MW) dwarf spheroidal galaxies (dSphs) and their
orbital pericenter distances poses a potential signature of self-interacting
dark matter (SIDM). In this work we investigate this possibility by analysing
the range of SIDM scattering cross section per unit mass, $\sigma/m_{\chi}$,
able to explain such anti-correlation. We simulate the orbital evolution of
dSphs subhaloes around the MW assuming an analytical form for the gravitational
potential, adopting the proper motions from the Gaia mission and including a
consistent characterization of gravitational tidal stripping. The evolution of
the subhaloes density profile is modelled using the gravothermal fluid
formalism, where DM particle collisions induce thermal conduction that depends
on $\sigma/m_{\chi}$. We find that models of dSphs, such as Carina and Fornax,
reproduce the observed central DM densities with fixed $\sigma/m_{\chi}$
ranging between $30$ and $50$ cm$^{2}$g$^{-1}$, whereas other dSphs prefer
larger values ranging between $70$ and $100$ cm$^{2}$g$^{-1}$. These cross
sections correlate with the average collision velocity of DM particles within
each subhalo's core, so that systems modelled with large cross sections have
lower collision velocities. We fit the cross section-velocity correlation with
a SIDM particle model, where a DM particle of mass $m_{\chi}=0.648\pm 0.154$
GeV interacts under the exchange of a light mediator of mass $m_{\phi}=0.636\pm
0.055$ MeV, with the self-interactions being described by a Yukawa potential.
The outcome is a cross section-velocity relation that explains the diverse DM
profiles of MW dSph satellites and is consistent with observational constraints
on larger scales.
