Paul,
By "fine" lines, do you mean a single tuft-line?
I inspected an axminster a few years ago which had been cleaned (to a fairly acceptable standard) then exhibited a similar symptom running through the pattern, but affecting the red fibres only. These 'lines' were about 9" or 10" apart.
I uplifted the carpet at one end and removed sample red pile from the part of the carpet which had been folded into the gully, so not cleaned. All the samples that were in line with the 'white lines' surrendered their dye readily when given a short soak in mild sodium bicarbonate solution, then pressed dry between tissues.
Red pile removed from between these lines, though showing some bleed, maintained close to the original colour.
My conclusion was that the mill had acquired a faulty dye batch and that these spools had been loaded on the loom at regular intervals. Within ten days of my report being handed to the retailer , the mill contacted the customer direct inviting her to visit the retailer to select a replacement carpet from their range, at their expense. My conclusion was neither confirmed nor denied, but the readiness to replace £2,500 without further inspection suggests that they were aware of a problem with some of their output.
The fact that we both refer to red fibres is no coincidence since this colour tends to be the most susceptible to bleed.