How to save a file in a selected directory in Flutter for Android

Question

I'm trying very hard to understand how to get the permission to write files in a selected directory with File Picker in Android. I'm also using permission handler package. The code is correct, I'm sure. It is something related to permissions in Android Manifest file (but I have already specified read and write external storage permissions).

The dialog in the app for allowing permissions doesn't appear. It directly opens me (as I intended in my code) the setting page to set manually the permission. But here, it is written that the app doesn't require any permission. If I try to write a file in the selected directory, the operation fails and the compiler says to me that this operation isn't allowed, something likes that. Thank you for any help.