Does this strategy work?

I'm thinking about the following strategy for Fastest way to collect an arbitrary army:

1. When a soldier decides to go to some house he "reserves" it.
2. Once a soldier is free (has delivered the news to another house) he takes the closest house from all unreserved and goes there. If there are several closest houses he chooses at random.

My question is "Is there a house placement example for which this strategy gives a time bigger than $2+\sqrt{2}$?".

It is easy to show that the soldiers will go consecutively from one group to another. So the path will take $2*0.25+2*0.5+2*0.75+3*1+\sqrt{0.5} = 6.7 > 2+\sqrt{2}$.