That's a super good question.
In fact, for that kind of models, we captured the main phenomenological consequences (from the dark matter standpoint) with our next-to-minimal setup with two mediators. I thus consider moving to other topics for what regards studie of composite models. I plan to still follow the path of non-minimality, but in order to explore the impact of new signals not covered by the experimental programme at the LHC. This will allow us to make sure the LHC is exploited in the best possible way.
Ergh.... I think that you just selected the next topic about which I will write (as I have results already published in that direction) :)
hahaha so the next steps are already well developed!
Well... that's probably true :D