ALDI Nord and Süd have some family issues and probably won't combine their US ventures. I see the Schwarz Gruppe (Lidl, Kaufland) as the biggest contender in the US. Kroger is probably gonna have a very bad time.
I didn't mean that they'd combine volumes, but both could fight for share. So you have 3 contenders, 2x Aldi and Lidl. Lidl probably has most money to spend but the US market is huge, they can all grow at their limits without threatening each other too much.