A Trader Joe's store slated to open up shop in the place of a shuttered supermarket at East 32nd Street and Third Avenue will open this summer, ahead of the previous projected opening date in the fall, according to Alison Mochizuki, a spokeswoman for the chain. Rumors have swirled online about a possible July opening, but Mochizuki did not offer any other specifics except that the store would open ahead of schedule.