Positive Burau eigenvalue conjecture for order-preserving braids

Let $B_n$ be the braid group on $n$ strands, let $\mathbb{E}$ be the relevant ordered field with quadratic form $Q$, and let

$$\rho:\beta \to \operatorname{GL}_{n-1}(\mathbb{Z}[t^{\pm 1}])$$

be the reduced Burau representation. A braid $\beta\in B_n$ is order-preserving if it preserves an ordering of the free or surface group under consideration. Positive Burau eigenvalue conjecture. If $\beta\in B_n$ is an order-preserving braid, then $\rho(\beta)$ has at least one positive eigenvalue in $(\mathbb{E},Q)$. This would provide a partial converse to the criterion that positive Burau eigenvalues imply order preservation; the full converse is indicated to be false by the examples discussed in the paper.