Let . Then by the usual estimate for infinite products.
Also, if m < n then But should lie between and , where That makes it look to me as though (T_n) should be a Cauchy sequence, which would therefore converge. Of course, you won't want the infinite product to diverge to 0, so you would also need to investigate whether is bounded away from 0 (presumably by ).
I don't pretend to have thought through this carefully, so I can't be sure that there is any mileage in this approach. Maybe you have already tried it. But it looks reasonably promising to me.