Only affecting "value" not "processed" property

In my JSON:API output of text fields using formats and filters, this rest_absolute_urls module seems to only be working on the "value" output string, not "processed", where the other necessary filters are applied.

How can I get this to work on the processed output, too?

Is it necessary to use a normalizer other than StringDataNormalizer like FieldItemNormalizer? Or does this require another input filter, like the way the rel_to_abs → module works?