That's a tough one to advise on w/o actually seeing it. I can tell you from years of experience when I see those deep reds in ANYTHING whether upholstery or area rugs I proceed with more caution. The reason is I would say that rug has a fairly high chance of bleeding and if it does it will almost certainly be the red dyes that will do it. I would not use the oxy buff shampoo or anything with oxy in the label for that matter just to be extra cautious on that particular rug. If you cannot determine for absolute certain the fiber content and there is no tag that is automatically good reason to ask the customer to sign a waiver. If they're not willing to do that pass on it, end of story. Cotton and rayon will burn similarly, but I haven't seen enough rayon/ viscose in my area to be really expert at identifying it. Wool is extremely easy to test because it smells exactly like burning hair. The fringes are almost certainly cotton and that may be the foundation of the rug. I strongly suspect the face fibers are not rayon, but wool, but I can't possibly be certain from a picture. Wool is a great fiber and normally very easy to clean. Rayon fiber will distort easily and if left distorted to dry it is near impossible to get it right again.
So, what to do. Be extra cautious how you clean it and especially what you use on it. I'd recommend either a good prespray that is wool and fine fabric safe or a wool shampoo. The natural sponge is what I use too on some rugs if I'm being over cautious. I also like to know the original cost of the rug before I clean it and again, no waiver on that and I'm not doing it, period. With a fine fabric shampoo followed by light extraction w/o getting too wet at any point that should clean up good. If it does turn out to be rayon you'll need to be extra careful about grooming it out as it dries. I would be surprised that the owner does not know what that rug is, but perhaps they just never paid attention. If it's an older rug that also lends to it NOT being rayon/viscose. Cleaning these kinds of rugs in the home can back fire on you and it would be advisable to take both a rug cleaning and upholstery cleaning course. Last thing, do whatever you have to in order to get it dry quickly. Bleeding happens as something is drying. If it dries with bleeding it's extremely difficult to correct. Those fringes don't look too bad and should respond well to shampoo. I don't know if the Oxy buff would harm it, but I'd be a little gun shy using it on that particular rug. Oh, one last very important thing. Be absolutely certain to check whatever chemical you use with a cotton towel for bleeding on that rug. I'd be surprised if at least the red doesn't test positive for bleed, but again, I'm looking at a picture. I hope this helps.