I just won my first limited competitive treasure hunt ever in Manhattan and I did it with only owning 1 property in the entire city.
I got a lucky spawn having come right out of the train station and was able to find the treasure in 19 seconds. I didn't even need to utilize the property that I had bought but I know that I will eventually.
I'm still looking to expand more in Manhattan, maybe try to eventually get to 4 properties total but for now it's nice to see that I can have success in a city even with owning a limited number of properties in the game.