Degree of decidable categoricity

Determine whether every degree of categoricity is a degree of decidable categoricity, including the stated conjectural range and the case of d.c.e. degrees.

Background

It is known that every degree of decidable categoricity is a degree of categoricity. The converse is conjectured for sufficiently high jumps, while the general and d.c.e. cases remain open.

References

Every degree of decidable categoricity is a degree of categoricity. Is the converse true?

A conjecture for this problem is that it is true for $d \geq 0{(\alpha)}$, where $\alpha$ is 'sufficiently large' computable ordinal. Say, for $\alpha = \omega$. What happens with d.c.e. degrees $d$?

— Open Problems in Mathematical Logic  (2608.26628 - Barmpalias et al., 27 Aug 2026) in 2025 Section 7