Just because you can purchase a code share seat on a flight, does not mean there is an award seat available on that flight.

I don't know when in June you are traveling, but there are saver seats on a connection flights from the 1st-12th. After that only a couple of days have a saver award available. Some of these connections are on a LH leg, if flying oh LH is important to you. Good Luck & Safe Travels!