We currently store the index name in the index for each document in order to be able to handle multi-index search. That's because multi-index search requires that we resolve, for each document, the index this document originated from.

This storage is a bit weird, because we will store the exact same value for every single document in the index.

We could probably derive this information from the Lucene components (LeafReaderContext/LeafReader). Let's do that instead, so that we can get rid of the weird storage.