Analytic realization conjecture for singular-crossing bimodules

The paper constructs bimodules $\mathcal{X}^{DA}$, $\mathcal{X}^{DD}$, and $\mathcal{X}_i^{DD}$, together with the Heegaard diagram in Figure~ and its variant globally extended to the left and right. The analytic bimodules are defined using techniques such as those in Ozsváth--Szabó's planned work, for suitable analytic choices. Analytic realization conjecture. The bimodules $\mathcal{X}^{DA}$ and $\mathcal{X}^{DD}$ constructed in this paper should be isomorphic to the $DA$ and $DD$ bimodules defined analytically from the Heegaard diagram, and $\mathcal{X}_i^{DD}$ should be isomorphic to the $DD$ bimodule of the globally extended variant, for some analytic choices. This would identify the combinatorially constructed bimodules with analytic Heegaard Floer bimodules and support their interpretation as invariants of singular crossings.