Two different issues here: whether the show is eligible for Tony nominations and whether it can actually get nominated.
In order to be eligible, the production has to invite the nominators, the Admin Committee, and the voters. I don't think it matters in the rules how many of them actually show up in terms of eligibility.
The production has to invite voters to at least 8 paid performances, including previews. If the production does not have 8 paid performances, it is still eligible if it invites people to at least 50% of the performances, but that number includes previews, so it's not as if High could invite everyone to the last four shows this week, because the show will have played 28 previews + 8 = 36 total paid performances.
So for High to be eligible, it would have had to have invited the voters to at least one performance before today's announcement. We don't know whether or not they did.
"What was the name of that cheese that I like?"
"you can't run away forever...but there's nothing wrong with getting a good head start"
"well I hope and I pray, that maybe someday, you'll walk in the room with my heart"