Lattice-cohomology morphism conjecture for deformations of isolated curve singularities

Let $\{(C_t,o)\}_{t\in(\mathbb C,0)}$ be a flat deformation of isolated curve singularities. For each level $n$, let $\ell^*$ denote a vertex of the level set of the generic fibre and let $\ell^*_{\lim}$ be its limiting lattice point on the special fibre. The map

$$\ell^*\mapsto \ell^*_{\lim}$$

from the $0$-skeleton of the generic level set to the $0$-skeleton of the special level set should lift to a map of the corresponding level sets

$$\widetilde X_{t\not=0}\cap S_n\to\widetilde X_{t=0}\cap S_n$$

that induces a degree-zero graded $\mathbb Z[U]$-module morphism

$$\mathbb H^q(C_{t=0},o)\to\mathbb H^q(C_{t\not=0},o)$$

for every $q\geq1$. Lattice-cohomology morphism conjecture. Such a lift and induced morphism exist for every flat deformation of isolated curve singularities. This is proposed as an enhancement of the paper's deformation theorem and would provide functorial maps on higher lattice cohomology under specialization; the source gives no resolution of the conjecture.