US pastor Brunson's lawyer, Ismail Cem Halavurt, spoke to Bloomberg by phone
So far, I believe only the lower court has rejected the earlier appeal and that we're still waiting on the higher court's decision. But it's almost certain that the appeal there will be rejected as well so this appears to be the next course of action.
However, the likelihood of any of this succeeding is pretty much nil. It's basically just for show. Once the appeal to the constitutional court is rejected, the next move will be to apply for the European Court of Human Rights.
Looking at ECHR protocol, hearing for some cases can take as long as up to a year and they are in no right to overrule national decisions nor are they to act as a court of appeal in relation to national courts. That basically means it isn't going to be of much help either.