Cause one: soil load
The first and more hopeful cause is simply accumulated soil concentrated where people walk. Grit, oil transferred from shoes and skin, and airborne particulate settle heaviest along the path of travel. This kind of greying responds well to cleaning, sometimes dramatically, because the material sitting on and between the fibres is removed and the original colour returns. If the lane lightens noticeably when you dampen a small area with a cloth, soil is likely the main issue.