Loquacious highness characterization of isomorphism completeness for classes of structures

Let $K$ be a class of structures. A degree is loquaciously high for isomorphism for $K$ if it computes, uniformly from any pair of structures in $K$, functions that are inverse isomorphisms whenever the structures are isomorphic. A degree is uniformly high for isomorphism for $K$ if it computes, uniformly from any pair of isomorphic structures in $K$, an isomorphism between them. The isomorphism problem for $K$ is the set of pairs of structures in $K$ that are isomorphic.

Loquacious highness characterization. If every degree loquaciously high for isomorphism for $K$ is uniformly high for isomorphism, then the isomorphism problem for $K$ is $\Sigma^1_1$-complete.

This is posed as an open problem concerning the relationship between structural highness notions and the complexity of isomorphism problems. The statement is related to the preceding results showing that, under $\Sigma^1_1$-completeness, loquacious highness implies uniform highness together with enumeration of Kleene's $\mathcal{O}$, but the converse characterization remains open.