From T171797#3478411:

There are three cases with the wrapclass parser option:

1. Wrapping with the default "mw-parser-output"
2. No wrapping at all
3. Wrapping with some non-default class

As far as actual parser output beyond the wrapper div itself, the only thing currently varying on this option is TemplateStyles. And TemplateStyles generates the same output for #1 and #2. So what we could probably do is:

• Deprecate #2 (the ability to set wrapclass to false for "no wrapping") in favor of an "unwrap" transformation that removes the default wrapper div.
• Change MessageCache/Message, TextExtracts, ParsoidBatchAPI, and MobileFrontend to use the "unwrap" transformation instead of setting false. Also update tests in TemplateStyles and Wikibase to do whatever is appropriate where they're currently setting false.
• Remove wrapclass from ParserOptions::$inCacheKey, causing ParserOptions with a non-default wrapclass to become non-cacheable.