Completions of triangulated categories without models

I will touch on my work (arxiv:2401.00130) where we found an obstruction to bounded t-structures using Neeman's completion theory for triangulated categories. I will touch on what happens when one looks at the known examples of triangulated categories without models (including the one constructed by Muro, Schwede, Strickland) although in some of those cases, it is easy to check that there are no bounded t-structures.