The genus 1 bridge number of satellite knots

Let $T$ be a satellite knot, link, or spatial graph in a 3-manifold $M$ that is either $S^3$ or a lens space. Let $\mathfrak{b}_0$ and $\mathfrak{b}_1$ denote genus 0 and genus 1 bridge number, respectively. Suppose that $T$ has a companion knot $K$ (necessarily not the unknot) and wrapping number $\omega$ with respect to $K$. When $K$ is not a torus knot, we show that $\mathfrak{b}_1(T)\geq \omega \mathfrak{b}_1(K)$. There are previously known counter-examples if $K$ is a torus knot. Along the way, we generalize and give a new proof of Schubert's result that $\mathfrak{b}_0(T) \geq \omega \mathfrak{b}_0(K)$. We also prove versions of the theorem applicable to when $T$ is a ``lensed satellite'' and when there is a torus separating components of $T$.