The one-unit gap conjecture for realizable dimensions

Let $G$ be a multigraph. Write $G^=$ for the multigraph obtained by adding the relevant parallel edges as defined in the paper, and let $\operatorname{rd}(G)$ denote its realizable dimension.

One-unit gap conjecture.

$$\operatorname{rd}(G^=)-1\leq \operatorname{rd}(G)\leq \operatorname{rd}(G^=).$$

The upper bound is already known from the definitions, while the conjectural content is that the difference between the two realizable dimensions is never more than one; the paper notes that no example with a larger difference is known.