The proper term for that conductor is the grounding electrode conductor (GEC). It is required to be protected if it is smaller then #6 AWG or if it’s subject to physical damage. If not in a raceway then it needs to be secured to the building surface.
250.64(B) Securing and Protection Against Physical Damage. Where exposed, a grounding electrode conductor or its enclosure shall be securely fastened to the surface on which it is carried.
Grounding electrode conductors shall be permitted to be installed on or through framing members.
(1) Not Exposed to Physical Damage. A 6 AWG or larger copper or aluminum grounding electrode conductor not exposed to physical damage shall be permitted to be run along the surface of the building construction without metal covering or protection.
(2) Exposed to Physical Damage. A 6 AWG or larger copper or aluminum grounding electrode conductor exposed to physical damage shall be protected in rigid metal conduit (RMC), intermediate metal conduit (IMC), rigid polyvinyl chloride conduit (PVC), reinforced thermosetting resin conduit Type
XW (RTRC-XW), electrical metallic tubing (EMT), or cable armor.