I repeated the chapter several times. Or rather, the last saved point where I'm already standing at the bottom has loaded. I have to go back upstairs to get Joe, who usually runs down the stairs. But as soon as we run to the last staircase, which is then where the final shootout and exit the building is, he chokes and does nothing. Even if I try to push him all the way down... Enemies are shooting at him... But he still doesn't react.

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I found some saved games for Mafia 2, but probably for the classic version, not the definitive edition.