Schleimer–Thompson conjecture on the complexity of fibered knots

Let $M$ be a three-manifold, and let $K\subset M$ be a fibered knot whose page is a surface $\Sigma$ with $\chi(\Sigma)\leq -3$. Let $d_{\mathcal{AC}}(\phi)$ denote the complexity of its monodromy $\phi:\Sigma\to\Sigma$, namely

$$d_{\mathcal{AC}}(\phi)=\min\{d_{\mathcal{AC}}(\gamma,\phi(\gamma))\mid \gamma\text{ is a vertex in }\mathcal{AC}(\Sigma)\}.$$

Schleimer–Thompson conjecture. For any three-manifold $M$, there is a constant $t(M)$ such that if $K\subset M$ is a fibered knot, then the monodromy of $K$ has complexity at most $t(M)$. Moreover,

$$t(S^3)=1.$$

The conjecture predicts a bound on the arc-and-curve-complexity of monodromies of fibered knots depending only on the ambient three-manifold, with the sharp value asserted for the three-sphere. Its status is not resolved in the supplied source context.