I apologise in advance because I'm obviously new both to causal inference and to this community.
I would like to estimate the impact of a treatment on firms' behavior. I'm considering using the synthetic control method (in particular, the relatively new synthetic DiD by Arkhangelsky et al., 2021) because I am concerned that the parallel trend assumption is not going to hold. Moreover, treatment adoption is staggered and the framework by Arkhangelsky et al. (2021) can be easily adjusted to account for it.
However, I am also concerned about self-selection into treatment for which it is common to use either some form of matching (propensity score, mainly) or propensity score weighting.
I haven't been able to find any work that uses the combination of the two. Mostly, I found that researchers make separate use of synthetic controls and DiD with propensity score weighting or matching to evaluate robustness. Are the two methods really theoretically and practically difficult to integrate?