Necessity of the uniform computability of constant functions in the main equivalence

Determine whether the equivalence "M has a decidable presentation if and only if M^{[0,1)} has a computable presentation" remains valid when the assumption that the constant functions in M^{[0,1)} are uniformly computable points is removed.

Background

The main theorem shows: a countable classical structure M has a decidable presentation if and only if its Borel randomization M^{[0,1)} has a computable presentation for which the constant functions are uniformly computable points.

The authors provide sufficient conditions (e.g., effective recognizability) under which the uniform computability assumption can be dropped, but they do not know whether the assumption is necessary in general.