Is there a clue in this sentence to know it should be "recognize/are getting to know" instead of "know/recognize"? When I saw the word I remembered recognize but I had to use are recognizing.
This verb can mean both. And therefore, both are accepted here.
The reason to prefer "getting to know" here is only because it makes more sense. But both are accepted here.
Why is "Those girls know those cities" incorrect?
Because the girls don't know the cities yet, they are still in the process of getting to know them (verb poznávat).
Your sentence would be "Tamta děvčata znají tamta města" (verb znát).