The reopening of Family History Centers will primarily be a local decision based on the guidelines that are under the direction of local authorities.
The timing for the opening of Local Family History Centers is under the direction of local leaders and is not controlled or scheduled by Salt Lake.
Just an additional piece of information.
"The decision to reopen a FHC is a local priesthood decision, specifically the decision is to be made by the Area Presidency."
@Nancy Elaine Watson
I do not know about anyone else ...
But, I certainly have not aware (ie. seen any updates from the Leaders of the Church) of any update for the opening of "Family History Centres" of the Church.
And, when they do open, such openings may be staged at different times, depending on where one lives in the World.
If you are a Member of the Church, you may wish to contact the local Leaders of your Stake/District. And/or, check out the "E-mail" News notifications from the Church.
If you are not a Member of the Church, you could check out the "News" from the Church "Website" (every now and then) at: https://www.churchofjesuschrist.org/?lang=eng
Particularly, News releases at: https://newsroom.churchofjesuschrist.org/news-releases/
Here is a "News" Article that is 'Dated' 3 June 2020.
As far as I could tell, in a quick look, no direct mention of (or, inference to) "Family History Centres" of the Church.
You could also keep an eye on the "Web" page for the "Family History Library" of the Church in Salt Lake City, Utah, USA - to see when the "Closed" Banner is removed.
I know that this does not help, just thought I mention a few avenues to keep checking out.
Unfortunately right now the restricted records are not accessible at the family history centers either. Not until the portal is up and going again.