L_o_G,
Chris is probably right that this is filtration soiling.
Ultrapac Renovate with good agitation and a minimum of 10 minutes dwell-time is effective, but I now use Eco Spotter which is more effective and does not require dwell.
The closer you get to the draught source, the more difficult the removal will become and as you get to the edge either a crevice tool or a local uplift may be required.
The customer should be made aware that the staining may not be completely removed and that it will start re-forming immediately unless the causal defect - the gap between skirting and floor - is sealed.
In some cases sealing this gap serves to encourage the draught to enter through the gaps between the floorboards instead, which can lead to dark line forming along these gaps or any un-sealed joints between sections of underlay.

The total solution is that the carpet be fitted fully in line with current British standards, which will never happen unless the carpet fitter is paid the appropriate fee.