Does it make any difference when the upgrade is done?
If you upgrade at the time of ticketing, so that the upgrade is included in the price, I would assume that you are ticketed in T - so that all you need is T inventory available on whatever flight you want to get on.
If, on the other hand, you ticket it in L, and then decide you want to upgrade to T on a particular flight, then I would assume that L would have to be available on that flight (so you can get on that flight) before you can then request to pay the extra to upgrade. Alternatively, you could have the ticket "re-issued" (+ whatever fees) to include that segment in T, at which point the availability of L inventory would no longer be relevant.
No idea whether any of that is, in fact, how it works, just supposition on my part.