Is WRITE permission needed to edit a file in the ASD?

This is not possible because the ASD is only accessible to the app with which it was created.
See here: Some basics on Android storage system

 /storage/emulated/0/Android/data/<packageName>/files/

which is in the External (private) Storage, but does not require READ_ / WRITE_EXTERNAL_STORAGE permissions. The ASD can only be accessed by your app and is automatically removed when the app is uninstalled.