Arc distance equals level number

Let $K$ be a knot in $1$-bridge position with respect to a genus-$g$ Heegaard surface that splits a $3$-manifold $M$ into two handlebodies $V$ and $W$. One can move $K$ by isotopy keeping $K\cap V$ in $V$ and $K\cap W$ in $W$ so that $K$ lies in a union of $n$ parallel genus-$g$ surfaces tubed together by $n-1$ straight tubes, and $K$ intersects each tube in two arcs connecting the ends. We prove that the minimum $n$ for which this is possible is equal to a Hempel-type distance invariant defined using the arc complex of the two-holed genus-$g$ surface.