I played Undying (the original one) in this current free arcade game. The mandatory condition is to kill him with the pen.
However, Diana in the opening statement suggests 3 ways to kill him: on the spot, with the sniper across the race track, or with the pen. I poisoned his coffee - failed. I sniped him across the race track - failed.
I think this is a bit misleading 
The intro videos in Arcade mode that specify a certain kill method are carried over from when they are live ETs. The Arcade missions spell out what needs to happen to succeed, regardless of what Diana’s intro states. Dont follow the intro video; follow the Arcade requirements.
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The second stage of the Arcade contract has no restrictions, you can try one of the other fun opportunities there!
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Does throwing the regular exploding pen at his head fail the mission on part 1?
When I tried it back when that ETA launched, it unfortunately did trigger an insta fail.