Hi Dan
Place rug, or at least the fringed area, on a mat or drop sheet. Pre spray well with chosen solution, I find a Micro-Splitting solution to be the best. Agitate with a hand brush with the fringe and across the "webbing" rinse well with an upholstery or low moisture tool. Extra drying strokes. If fringes are white, they will be slightly greyish whilst damp but dry better. They can also be bleached but this may lead to fibre damage.
Safe and happy cleaning:)
Ken