I am currently reading an article titled "Embedding and Coding Below a 1-Generic Degree" by Greenberg and Montalbán(link to a free source:https://pi.math.cornell.edu/~erlkonig/Papers/below1generic.pdf). At the end of section 3 in the paper they mention a few questions left without answer, one of them being: Is there a 1-generic degree g such that Th(D(< g)) is more complicated than true arithmetic? Where Th(D(< g)) is the theory of arithmetical degrees below the degree of the 1-generic.

My question is, has there been any resolution/progress on this question given that it was posed 20 years ago?

I am currently reading an article titled "Embedding and Coding Below a 1-Generic Degree" by Greenberg and Montalbán(link to a free source:https://pi.math.cornell.edu/~erlkonig/Papers/below1generic.pdf). At the end of section 3 in the paper they mention a few questions left without answer, one of them being: Is there a 1-generic degree g such that Th(D(< g)) is more complicated than true arithmetic? Where Th(D(< g)) is the theory of turing degrees below the degree of the 1-generic.

My question is, has there been any resolution/progress on this question given that it was posed 20 years ago?