Nightmare Realm and fleeing

There Comes a Reckoning card, Nightmare Realm.
Shootout Hex 3, Boot: Choose an opposing dude. That dude cannot leave the shootout this round. Each round before shooters are declared, give that dude -1 bullets and -1 value. Give your dude +1 bullets.

as I understand from the shootout timing structure, the “run or gun” portion is at the end of the first round of the shootout, and thus the targeted opposing dude would not have the option to flee, correct? (Leaving the shootout with a Smart Choice or Personal Ornithopter are obviously No Go, but the fleeing is a little more questionable.)

Correct!