Current status of the LQG graviton propagator:In LQG there are NO gravitons at the fundamental level.
You know the story at least as well as I do. The graviton is a mathematical concept primarily at home in a stable flat geometric setting. High energy means curved unstable geometry. No question of gravitons being "fundamental", in the usual sense of a high energy regime. The graviton propagator for then-current LQG was derived around 2006, then a new spinfoam vertex got established around 2008 and the calculations are being repeated.
Corrections welcome if anyone has more up-to-date information about the LQG, or a better way to summarize the situation.