We call a knot $K$ in a 3-manifold $M$ fibered if $M\backslash K$ fibers over $S^1$ with fibers $\Sigma$ and such that $K$ is ambient isotopic to the boundary of the compactified fiber $\overline{\Sigma}$. Given a fibered knot $K$ of genus $g$, by gluing two copies of $\overline{\Sigma}$ together we get a Heegaard surface for $M$ of genus $2g$.

Is it known whether the difference between the Heegaard genus of $M$ and the minimum of twice the genus of a fibered knot in $M$ can be arbitrarily large?

In general, are there techniques besides Heegaard genus to bound from below the minimal genus of a fibered knot in a 3-manifold? I know that Ken Baker has worked on counting genus one fibered knots in lens spaces, but I don't know of any other work in this area.