No the resulting T would not react immediately. This doesn't happen in existing D-T reactors, why would it happen here where the T/D ratio orders of magnitude lower? Most atoms in a cycle will never fuse.
My reasoning is that, at plausible temperatures, D-D fusion is say 5x more frequent than D-He3 reactions, and then D-T fusion is ~1000x more reactive than D-He3.
So if 0.1% of the He3 fuses, then ~0.5% of the sample will become tritium, most of which will fuse.
Either I'm significantly overestimating the amount of reactants that are expected to fuse in each cycle, underestimating their target temperature, or they plan on have a very helium-rich mix of fuel.