Häggkvist's conjecture on compatible Hamilton cycles in Dirac graphs

Let $G$ be a Dirac graph, meaning a graph on $n$ vertices with minimum degree at least $n/2$. Let $\mathcal{F}$ be an incompatibility system over $G$, consisting for each vertex $v$ of a family of unordered pairs of distinct edges incident with $v$; edges in one of these pairs are incompatible. The system is 1-bounded if every edge incident with any vertex is incompatible with at most one other edge at that vertex. A Hamilton cycle is compatible with $\mathcal{F}$ if every pair of its edges is compatible.

Häggkvist's conjecture. For every 1-bounded incompatibility system $\mathcal{F}$ over a Dirac graph $G$, there exists a Hamilton cycle compatible with $\mathcal{F}$.

The paper presents this as a conjecture of Häggkvist from 1988 motivating its study. The supplied text gives no resolution status, so the conjecture is recorded as open.