It seems Private Browsing makes localStorage read only:
http://spin.atomicobject.com/2013/01/23/ios-private-browsing-localstorage/

When loading store.js, I get this error:
QUOTA_EXCEEDED_ERR: DOM Exception 22: An attempt was made to add something to storage that exceeded the quota.

I saw these lines in the lib:

try {
  store.set(namespace, namespace)
  if (store.get(namespace) != namespace) { store.disabled = true }
  store.remove(namespace)
} catch(e) {
  store.disabled = true
}

Maybe there's something to improve there…