If you take Volkov with you to the arena and he gets damaged he will try to dock to repair his armor. Apparently this is not possible and he will remain for ever in the arena trying to repair his armor. The only thing you can still get from him is "Repairs almost finished" when you ask for his status.
This is in 1.7b2

watch tv, do nothing 16 Nov 2016:

Wingmen make no sense in the arena from both mechanical and lore perspectives. They should bypass the Rigel system entirely. You should get radio messages when you go through a Rigel gate from wingmen telling you that they will "meet you in the next system" and simply not spawn non-auton wingmen in Rigel at all.

Alternately wingman could dock while in Rigel and refuse commands, stating that they prefer to watch the action or have no stomach for it; or giving some other excuse.

Or you could urge them to fight in your stead, take control of their ships during the battles, and have THEM become legendary champion. I bet Jenna at least would like that.

ferdinand 17 Nov 2016:

Right now, you must order wingmen to wait to prevent them following you into the arena. Since the arena changed it is much easier for them to follow you inside. I like the idea of wingman watching the fight.

nms 15 Aug 2017:

This is resolved in 1.8a2, though I introduced a more common problem which will be resolved in 1.8a3.

nms 25 May 2018:

This, and the bug I introduced with wingmen ignoring orders, are resolved.