Constancy of analytic lattice cohomology in $p_g$-constant deformations

Let $(X_t,o_t)$ be a flat deformation of normal surface singularities with rational homology sphere links, and suppose that the geometric genus $p_g$ is constant along the deformation. Denote the degree-zero analytic lattice cohomology by $\mathbb H^*_{an,0}(X_t,o_t)$.

Constancy conjecture. The graded $\mathbb Z[U]$-module $\mathbb H^*_{an,0}(X_t,o_t)$ is constant along the deformation.

This would provide deformation invariance of the degree-zero analytic lattice cohomology under flat $p_g$-constant deformations. The source formulates it as a conjecture and gives examples supporting it; no general proof or disproof is stated.