I'd have to say, I've not seen any fresh closures on it. Not been down it for a couple of months, but I usually see what conditions are like before I go.
I would say though, regardless of which wheels/tyres you have on, you should stay on piste anyway, otherwise you just add fuel to the anti's fire. Also worth noting, and I don't know if this really happens, but the farmer has signs up stating anyone seen causing damage will be reported. May just be a little something to scare people, but I personally don't think its worth the risk.
If you fancy it, there's a rather interesting lane near Wellesbourne, we could always try and get a decent route sorted and make a day of it?