Announcing that the great tribulation has begun would be a bold move. I don't think they have it in them, but desperation can make people very bold, indeed.
But wouldn't there need to be definitive 'signs' to indicate that the GT was underway? The name itself implies that something notable and terrible is happening. Would it be enough to point at the "world situation"? That's something they do all the time as a matter of habit, why would the current situation be different?
On the bright side, this might be the sort of thing that leads to a larger-than-normal exodus of followers. And in that sense, it would be a self-imposed great tribulation for the WTS.