Outdoors maybe. Sadly not "open air"
Section 193 Law of Property Act 1925. Public has a legal right to take "air and exercise" on Urban Common land such as Mynydd Llangattock which includes Agen Allwedd and Daren Cilau, and the Ogof Draenen catchment. Note this access right is not restricted to open air.
NRW accepts that caving is air and exercise. They'd have to after DEFRA lost the Billson Judicial Review in which it was argued by DEFRA that horse riding isn't air and exercise.
So NRW goes on to say that urban commons don't include the caves in the land. This is despite Section 205 defining "land" as including mines and minerals, any strata or seam of minerals or substances in or under any land. In other words, land is three dimensional not just the surface.
Make your own mind up.
There is hope, in Wales anyway. It seems the Welsh Government's preferred and more inclusive term to replace "air and exercise" (LPA) and "open-air recreation" (CROW) and "exercise and recreation" (Forestry Act) is simply "non-motorised recreation".
I hope the WG will eventually amend the CROW Act to differentiate Wales from England by saying that the public access right in Wales will be for "non-motorised recreation" while leaving caving off the "banned list" at Schedule 2 which is the status quo.