There's no telling, but even if it just gets posted on June 10 you shouldn't have trouble getting seats. Note that many problems have been reported with completing reservations on the site. Tell your credit card issuer that you'll be making foreign charges and make one attempt. If you'll be in Italy much sooner, you should be able to reserve once you're there. The day before is usually fine, but if you're taking a night train it's good to reserve a little earlier.