Harrington et al.'s unbounded exclusive-sum-number gap conjecture

Let $\epsilon(G)$ denote the exclusive sum number of a graph: the least $r$ such that $G$ together with $r$ isolated vertices has an exclusive sum labelling, meaning that the only working vertices are the added isolated vertices. Let $\operatorname{sum}(G)$ be the sum index. Harrington et al.'s conjecture. For every positive integer $N$, there exists a graph $G$ such that

$$\epsilon(G)-\operatorname{sum}(G)>N.$$

The conjecture asserts that the known upper bound of the sum index by the exclusive sum number can have an arbitrarily large gap. The paper says that it gives an explicit construction establishing this assertion.