Analytic lattice cohomology as a submodule of topological lattice cohomology

Fix a topological type, with associated topological lattice cohomology $\mathbb H^*_{top,0}(M)$, and let $(X,o)$ be any analytic type supported on it. Let $\mathfrak{R}_{top,0}(M)$ and $\mathfrak{R}_{an,0}(X,o)$ denote the corresponding graded roots.

Analytic lattice cohomology conjecture. The morphism

$$\mathbb H^*_{an,0}(X,o)\to \mathbb H^*_{top,0}(M)$$

is injective. At the level of graded roots, the graded graph morphism

$$\mathfrak{R}_{top,0}(M)\to \mathfrak{R}_{an,0}(X,o)$$

is surjective on vertices and edges. Equivalently, the analytic lattice cohomologies arising from analytic types supported on the fixed topological type should be precisely characterized among the graded $\mathbb Z[U]$-submodules of $\mathbb H^*_{top,0}(M)$.

This conjecture makes the preceding classification problem more precise: it seeks to determine which analytic lattice cohomology modules can occur for a fixed topological type. The source gives examples supporting it, but no resolution is stated.