Holomorphic potential conjecture under the operator conjecture

Let $\nu_\kappa$ be the measure on $\mathcal A$, let $E\subset\mathcal A$, and let $\mathcal A^\omega$ be a subset of full $\nu_\kappa$-measure. Let $\mathscr S$ and $\hat{\mathscr S}$ be the functions appearing in the operator construction.

Holomorphic potential conjecture. Assuming the limiting-measure and operator conjecture, there exists a set of full $\nu_\kappa$-measure

$$\mathcal A^\omega\subset E\subset\mathcal A$$

and a holomorphic function $\Xi$ on $E$ such that

$$\operatorname{Re}(\Xi)=\hat{\mathscr S}-\mathscr S$$

on $\mathcal A^\omega$.

This assertion supplies a holomorphic potential for the difference between the two functions governing the adjointness problem. Because it is conditional on the preceding conjecture and no resolution is given, it remains open.