The degree-spectrum characterization of successor recoverability for computable block functions

Let $f$ be a computable block function on $(\omega,<)$, and consider the degree spectrum of $f$ on $(\omega,<)$. The successor is recoverable from $f$ on $(\omega,<)$ when it can be computed from $f$ in the relevant presentation.

Degree-spectrum conjecture. If the degree spectrum of $f$ on $(\omega,<)$ is equal to all c.e. degrees, then the successor is recoverable from $f$ on $(\omega,<)$.

This would extend the equivalence between the c.e.-degree-spectrum property and successor recoverability beyond the classes covered by the paper's main theorem, to all computable block functions. The source does not state a resolution, so the conjecture remains open.