Halo Infinite's unannounced battle royale game has been mentioned by a former Certain Affinity developer.

343 Industries' original vision for Halo Infinite was an interesting one. Both the F2P multiplayer and premium campaign came to fruition, but the team was also working on a new gametype alongside longtime franchise partner Certain Affinity.
The mode was codenamed Tatanka, and would've been Halo's take on the then-popular battle royale craze (for reference, Call of Duty Warzone, a breakout battle royale hit, achieved 85 million players in less than a year and contributed many millions of mTX revenues).
Tatanka was never officially confirmed or announced by either team, but it's been said for a while now that the BR mode has been cancelled. Now we have the first real discussions about Halo Infinite's battle royale, courtesy of former Certain Affinity game director Mike Clopper.
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On LinkedIn, the former director says that Tatanka would've been a "game changer" for the Halo franchise:
"MP Halo Infinite - I led a large team of designers working on a canceled Battle Royale mode for Halo. I believe this product could have been a game changer for the franchise. We loved playing it and working on it was a fantastic experience in spite of its cancelation."
Now that 343i has rebranded to Halo Studios, and moved from Slipspace to Unreal Engine 5, it seems highly unlikely that Halo Infinite will get any new modes like this in the near or even mid-range future.
Halo Studios has confirmed that multiple Halo projects are in development in UE5, and there's speculation that a Halo Combat Evolved remake in Unreal Engine 5 could be in the works.