Higher-order convex orders under free multiplicative convolution

We prove that free multiplicative convolution with a probability measure on $[0,\mathrm{\infty })$ preserves the $k$-convex order for every integer $k\ge 2$, assuming finite $(k-1)$st absolute moments for the two measures being compared and for the convolving measure. The compared measures may have arbitrary support in $\mathbb{R}$.