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I run along that bit quite often. Probably see a ranger on a horse about once every 10 runs.
(It's not reserved for horses, it's just no cycling because it's not wide enough, too steep in places and not a reliable surface. It'd be horrible if everyone cycled along it. Luckily most of the year it's too boggy to use on a bike.)
As an alternative I'd probably look in to using Chohole Gate in Richmond Park (by the Golf Club) that will then take you on to Robin Hood Gate. Either by using the top part of the grim cycle path and then going under the A3 using the Subway part way down, or by avoiding the A3 completely and going through the back of the estate in Roehampton all the way down Alton Road or Bessborough Road. (But it looks like the path to get from Alton Road to Norstead Place is signed as No Cycling.)
Greenbank-
Sort of see what you mean, there are a couple of steep bits but I don't think that many people would use it (certainly not in the winter slop). It's wide enough for a 4×4 most of the way. It would just be nice to not feel guilty when riding along what can be the nicest part of my commute.
I don't like signs, but a couple of "cycle consideratly - pedestrians, dogs and horses have priority" might help appease walkers and dissuade people from cycling like a nob.
Did not know there was a gate by the golf course. Will go find it tonight!
Scrabble
Good map. It boils my piss that you're not supposed to cycle by the cemetery. I do regualrly, but would rather not feel guilty. The alternative is the grimest cycle path alongside the A3.
I've been tutted at a few times but only stopped once. This was by a ranger on a horse - the only horse I've ever seen using this section it is supposedly reserved for!