Your question of Apple supporting or not supporting fitness + is not what is up for debate.
Its the fact that other people with Lightning HDMI adapters intending to put iPhone and iPad content on big screens now can’t 100% of the time. And according to Apple’s website, the product description says the following:
“Overview
Use the Lightning Digital AV Adapter with your iPhone, iPad, or iPod with Lightning connector. The Lightning Digital AV Adapter supports mirroring of what is displayed on your device screen — including apps, presentations, websites, slideshows, and more — to your HDMI-equipped TV, display, projector, or other compatible display in up to 1080p HD.
It also outputs video content — movies, TV shows, captured video — to your big screen in up to 1080p HD. Simply attach the Lightning Digital AV Adapter to the Lightning connector on your device and then to your TV or projector via an HDMI cable (sold separately).