For a countable structure $\mathcal{S}$, let the *cospectrum* of $\mathcal{S}$ be the set $CS(\mathcal{S})$ of reals (non-uniformly) computable in every copy of $\mathcal{S}$ *(we can also make sense of cospectra for uncountable structures, via forcing)*.

The cospectrum is clearly an invariant with respect to Muchnik reducibility *(where the mass problem corresponding to a countable structure is the set of copies of the structure)*. Richter showed that in many cases the cospectrum is a silly invariant; for example, any two linear orders have the same cospectra (namely the set of computable reals). However, "higher-order" cospectra turn out to have more content. Namely, say that a structure $\mathcal{S}$ is:

*self-escaping*if for every copy $S$ of $\mathcal{S}$ there is some $f\le_TS$ such that $f$ escapes every $g\in CS(\mathcal{S}$.*self-dominating*if for every copy $S$ of $\mathcal{S}$ there is some $f\le_TS$ such that $f$ dominates every $g\in CS(\mathcal{S}$.

It looks plausible that self-sufficiency considerations could be used to provide interesting separations with respect to Muchnik reducibility (and variants of self-sufficiency could be useful for other reducibilities); see e.g. Downey/Greenberg/Miller. However, I believe there is still very little known about this. In particular, I can't seem to separate the two kinds of "self-sufficiency" above *(which seems a good test problem: if it's truly hard, then this is probably a silly notion to consider)*.

Question: Is there a structure which is self-$\mathfrak{E}$-sufficient but not self-$\mathfrak{D}$-sufficient?