Convexity for twisted conjugation

Let $G$ be a compact, simply connected Lie group. If ${\mathcal{C}}_1,{\mathcal{C}}_2$ are two $G$-conjugacy classes, then the set of elements in $G$ that can be written as products $g=g_1{g}_2$ of elements $g_i\in $\mathcal{C}$_i$ is invariant under conjugation, and its image under the quotient map $G\to G\mathrm{/}\mathrm{Ad}(G)$ is a convex polytope inside the Weyl alcove. In this note, we will prove an analogous statement for twisted conjugations relative to group automorphisms. The result will be obtained as a special case of a convexity theorem for group-valued moment maps which are equivariant with respect to the twisted conjugation action.