The trunk number of satellite knots and Thurston norm

Assume $J \subset\mathbb{R}^3$ is a non-trivial knot, and assume $\hat k\subset S^1\times D^2$ is a satellite pattern. Let $N$ be the generalized Thurston norm of the homology class of the meridian disk in $S^1\times D^2$ with respect to $\hat k$. Let $K$ be the satellite knot of $J$ with pattern $\hat k$. We show that the trunk number of $K$ is strictly greater than $N$ times the trunk number of $J$.