A couple of years ago I would have said that Betelgeuse is not likely to explode in the Current Era of OA. But now I'm not so sure, since it has been dimming and brightening significantly recently.
However Betelgeuse is a relatively close star, compared to the size of the Terragen Sphere and the expansion of the OA civilisation, so it would be reached relatively early. Looking at the expansion chart, it was probably reached sometime in the 3500's AT, maybe a bit later because of practical and historical reasons. Efforts to stabilise the star would have started almost immediately, consisting of the mining and removal of the loosely bound and thin outer layers, and this could have sufficed to slow down activity in the star until the lifespan of the star has been extended significantly.