Finite-certificate conjecture for relaxation complexity in dimension three

Let $X \subseteq \mathds{Z}^3$ be a full-dimensional lattice-convex set, and let $\operatorname{rc}(X,Y)$ denote the relaxation complexity of $X$ relative to a finite set $Y \subseteq \mathds{Z}^3 \setminus X$.

Finite-certificate conjecture. There exists a finite set $Y \subseteq \mathds{Z}^3 \setminus X$ such that

$$\operatorname{rc}(X)=\operatorname{rc}(X,Y).$$

The preceding result gives a lower bound for the relaxation complexity of full-dimensional lattice-convex sets in dimension three, but does not provide a finite certificate for its exact value. This conjecture asserts that such a finite certificate always exists in dimension three.