That's pretty much what happened. Trouble was the stakes were already high, if Starlin didn't yield to a bluff (had one been pursued) it would have been an embarrassing backdown and a show of weakness on part of the allies.

"Bluff" because no one other than perhaps Stalin would have followed through—to essentially extend WWII (this time between former allies).

So pragmatism was the order of the day, thus Poland and EE were sacrificed.