The discount fares for June are probably not loaded in the system yet. This usually happens about 2 months in advance.

However since the soccer World Cup starts in Japan on June 1st, and lasts for a full month, I fear this year's June prices for flights to Japan won't be cheap.

Perhaps playing in your favor is the fact the US team starts off in Korea, and is not considered a favorite to make it to the ensuing rounds, some of which will be in Japan.