Bernhart–Kainen conjecture on dispersable regular bipartite graphs

Let $G$ be a bipartite graph that is $k$-regular. Its dispersable book thickness $dbt(G)$ is the minimum number of pages in a dispersable book embedding, and a graph is dispersable when $dbt(G)=\Delta(G)$, where $\Delta(G)$ is its maximum degree. Bernhart–Kainen conjecture. Every $k$-regular bipartite graph $G$ is dispersable, that is,

$$dbt(G)=k.$$

The conjecture is false for $k=3$ and $k=4$: the Gray graph has dispersable book thickness four, and the Folkman graph has dispersable book thickness five. It remains disproved as a general statement despite holding for $k\leq 2$.