I'm not sure this announcement means that the Tonys are not cancelled eventually. It depends on when Broadway re-opens, and I don't think that date is going to be decided for several more weeks. I would say it's unlikely the Tonys will be cancelled, though, because there's little advantage to that from the marketing angle. There have been award-worthy productions, performances, and other elements that would stand up against prior winners.
There were 16 shows that were supposed to open before the end of the season. I believe 5 of them have already announced closure or postponement. I would guess How I Learned to Drive, Take Me Out, and Sing Street would be added to that list.
Of the eight remaining shows, Six is obviously ready to go. Assuming schedules work out, Lehman and Plaza Suite are well-sold and probably ready to go as well. Company is a transfer, it's doing well at the box office, and the cast probably have at least a six-month contract, so it is probably going to happen. Diana, Mrs Doubtfire, The Minutes, and American Buffalo are the question marks IMO.