No, it won't be available until four weeks after Heavensward is released.

Check out the summary of the Live Letter XX here. The section you're looking for is under the "Release Schedule" heading on the second post.

Release Schedule

Everything that we discussed today will be available at the start of Heavensward. However, as I discussed previously, if we were to release items and content that have weekly restrictions, we are worried that it would create situations where players would try to rush through everything as soon as possible and force others to skip cutscenes. This would also lead to situations where players would not be allowed to level up and enjoy the new content at their own pace.

As such, we will be introducing the Alexander raid two weeks after the release of Heavensward on June 23. Unlike the Coil of Bahamut, players will be able to re-enter a floor in Alexander even after clearing it (similar to the Crystal Tower). You won't be able to obtain another item once you've received an item, but you will still be able to help your friends. This version is intended for everyone to enjoy the scenario surrounding the new raid.

Alexander (Savage) and the Allagan tomestones of Esoterics will become available two weeks after the release of the normal mode of Alexander. We have removed content with weekly restrictions for the first four weeks of release. Also, unlike previous item level caps, we will have more phases and the range of item levels is much greater than before. We would like players to spend their first four weeks enjoying the game freely, and discuss with other players to plan and decide on what jobs they will be playing following this schedule.

Additionally, we’re planning to release the new Frontline map at the same time Alexander (Savage) is released. Honestly, we were not sure if we should release this at the beginning of Heavensward, but we assume most players will be focusing on the main story and leveling, and this may cause a situation where it’ll take longer to get into a match, therefore, we've moved the release timing to this point.