  We present improved results of the measurement of the correlation between
galaxies and the intergalactic medium (IGM) transmission at the end of
reionisation. We have gathered a sample of $13$ spectroscopically confirmed
Lyman-break galaxies (LBGs) and $21$ Lyman-$\alpha$ emitters (LAEs) at angular
separations $20'' \lesssim \theta \lesssim 10'$ ($\sim 0.1-4$ pMpc at $z\sim
6$) from the sightlines to $8$ background $z\gtrsim 6$ quasars. We report for
the first time the detection of an excess of Lyman-$\alpha$ transmission spikes
at $\sim 10-60$ cMpc from LAEs ($3.2\sigma$) and LBGs ($1.9\sigma$). We
interpret the data with an improved model of the galaxy-Lyman-$\alpha$
transmission and two-point cross-correlations which includes the enhanced
photoionisation due to clustered faint sources, enhanced gas densities around
the central bright objects and spatial variations of the mean free path. The
observed LAE(LBG)-Lyman-$\alpha$ transmission spike two-point cross-correlation
function (2PCCF) constrains the luminosity-averaged escape fraction of all
galaxies contributing to reionisation to $\langle f_{\rm esc} \rangle_{M_{\rm
UV}<-12} = 0.14_{-0.05}^{+0.28}\,(0.23_{-0.12}^{+0.46})$. We investigate if the
2PCCF measurement can determine whether bright or faint galaxies are the
dominant contributors to reionisation. Our results show that a contribution
from faint galaxies ($M_{\rm UV} > -20 \, (2\sigma)$) is necessary to reproduce
the observed 2PCCF and that reionisation might be driven by different
sub-populations around LBGs and LAEs at $z\sim 6$.
