Early June has been very quiet for most of the 2010s, the exception being 2010 itself.
I definitely think the latter part of next week will have a chance beyond the amplification of the ULL near the Great Lakes. There is a strong signal in the GEFS for at least some activity in that period (and there has been for awhile), and note that the GEFS SCP chiclets on Gensini's website are anomaly based, rather than using raw count, so it gives a better idea of elevated activity relative to climo. Signal on the EPS is rather nebulous, but I wouldn't expect a really large signal in the heights at this point since it looks to be more centered around subtle disturbances in the flow.