Determination of the graded root by lattice cohomology for curve singularities

Let $(C,o)$ be an isolated curve singularity. Its graded root is the weighted rooted graph $\big(R(C,o),\chi\big)$ whose vertices of weight $n$ correspond to the connected components of the associated level set of weight $n$. The lattice cohomology module of $(C,o)$ is denoted by $\mathbb{H}^0(C,o)$.

Graded-root determination conjecture. The graded root $\big(R(C,o),\chi\big)$ is determined by the lattice cohomology module $\mathbb{H}^0(C,o)$.

For irreducible plane curve singularities, the lattice cohomology module determines the embedded topological type and therefore the graded root. The claim proposes that this determination remains valid for arbitrary isolated curve singularities, despite examples showing that the graded root can contain strictly more information than its associated $\mathbb{Z}[U]$-module; no counterexamples are known.