Schleimer's bounded translation distance conjecture for fibered knots

Let $M$ be a closed connected oriented $3$-manifold, and let $K\subset M$ be a fibered knot with monodromy acting on the arc complex of its fiber. The translation distance is the minimum distance between an arc and its image under the monodromy.

Schleimer's conjecture. There is a constant $t(M)$ such that the monodromy of every fibered knot $K\subset M$ has translation distance in the arc complex of the fiber at most $t(M)$. Furthermore,

$$t(S^3)=2.$$

The conjecture predicts a uniform bound depending only on the ambient closed $3$-manifold, with the sharp proposed value for $S^3$. The paper notes that the restriction to knots, rather than links, is essential, and proves that infinitely many fibered Montesinos knots satisfy the asserted $S^3$ bound.