Dobrinen–Simpson conjecture on high degrees and almost everywhere domination

Let $A$ be a set of natural numbers, let $A'$ denote its Turing jump, and let $0”$ denote the second Turing jump of the computable degree. A set is almost everywhere dominating when it has the almost-everywhere domination property discussed in the paper. Dobrinen–Simpson conjecture.

$$A' \geq_T 0” \Leftrightarrow A\text{ is a.e.~dominating}.$$

This conjecture asks whether highness is equivalent to almost everywhere domination. The surrounding text states that the reverse implication and the question of whether every high degree is almost everywhere dominating were not settled there; its resolution is therefore not established by the supplied source.