That would be very logical indeed, otherwise bridleways would not be connected.

The current version of this path contains an explicit mention about access tags here:
https://www.openstreetmap.org/way/1046461518/history/2
Corrected permissions. Signed for ONLY walkers. Bikes and horses not permitted.
Edited 12 months ago by Der Bierpuncheur
Changeset #133267104

Maybe that user was wrong.  That is a possibility.

On existing Mapillary pictures, the closest I could find is this, with a small path going uphill, and if you zoom a bit you can see a pedestrian logo on the post and nothing else. 
https://www.mapillary.com/app/?pKey=416798506854895

Maybe the situation has changed, or the middle part of the path has different access permissions than the first one.

@coleco: Can you consider surveying it yourself and bring the answers to this?  (I would love to do it too but I still haven’t mapped the thousands pictures or hours of videos from the last months, so I won’t take more.)