Hurlbert's ordering conjecture for universal cycles of permutations

Let $S_n$ be the set of permutations of $[n]$, and let $\mathcal{O}(n)$ be the clustered graph of overlapping $n$-permutations: its edges are the permutations in $S_n$, and consecutive permutations overlap in their last and first $n-1$ entries up to reduction. Hurlbert's ordering conjecture. Given any Euler tour of the clustered graph of overlapping permutations (that is, an ordering of $S_n$ such that each consecutive pair of permutations overlaps), there is a universal cycle for permutations that covers $S_n$ in the same order. Universal cycles correspond to Euler tours in this graph, but it remains open whether every such ordering can be realized by a universal cycle.