Just being picky here about terminology. A ledger rig is any rig held on the bottom by a lead weight. The term comes from coarse fishing where they also use floating rigs. Type "running ledger rig" into Google and you get 40K hits, including one from Tackle Tactics. For whatever reason fishos in NZ believe a ledger rig only describes one that has a sinker at the end and traces sticking out above. In other words, it's distinct from a running rig.
Anyway, to the original question. I don't know the answer. However, if a running rig or pulley rig is attached to a grip sinker and you tighten the line to put a nice bend in the rod tip, it won't run. If you feed out slack line, chances are a side current will put a bow in the line so it tightens and still doesn't run. Or the slack line drops into the waves and gets smashed about and loaded with weed.
I use pulley rigs because they're easy to make and use, but I don't kid myself they're fooling fish with their running abilities. Carp fishos know how to set up a running rig that actually runs with no tension felt by the fish. They're fishing slack water, which is a huge help, but they also lay their rods horizontally on a rack and use an electronic alarm to detect line movement. It's rarely practical to replicate that when surfcasting, although maybe you could do it in an estuary or in the Marlborough Sounds.