The bridleway at https://www.openstreetmap.org/way/378204098 isn't tagged with "designation=public_bridleway" but to the east https://www.openstreetmap.org/way/378204095 is and to the west https://www.openstreetmap.org/way/54276166 is, so it seems very likely this should be too.  This would benefit from resurveying.