Google, Wikipedia, and my paper maps, all classify the road from (near) Gullfoss to the ring road between Blönduós and Varmahlíð as an F-road (F35), which requires a 4WD and on which "normal" rental cars are not permitted (due to insurance reasons). However, Openstreetmap and more importantly Vegagerðin describe it as a regular road 35 with no F-prefix, and the Kerlingarfjöll tourist destination states that it is with good conditions accessible in all types of cars (despite Vegagerðin still considering the last 10 km on route F347 to be a mountain road). Openstreetmap tags state ref=35, old_ref=F35. Google Streetview indicates it is F35, but dates from September 2013. Elsewhere I've read that this road has no more unbridged river crossings.
Is route (F)35, Kjalvegur, over Kjölur, presently classified as a mountain road or a regular road, between Gullfoss and the ring road between Blönduós and Varmahlíð? Are regular rental cars permitted to travel on this route? Has this changed recently?