No, aggressive edge caching via IBM Cloud Internet Services (CIS) cannot interfere with, wipe, or desync HTML5 localStorage states.
localStorage is completely isolated within the user's physical web browser client and is strictly bound to the Same-Origin Policy (Protocol, Domain, and Port). Edge caching operates entirely on the network layer between the client and IBM Cloud Object Storage (COS); it does not possess the technical capability to clear or mutate data stored inside a user's browser sandboxed storage.
I belive this was answered here:
How Edge functions work
Caching