The group-action decomposition conjecture for invariant spaces of continuous functions

Let $G$ be a compact group acting continuously and transitively on a compact Hausdorff space $X$. A space of continuous functions on $X$ is called minimal if it is closed, invariant under the group action, and has no proper closed invariant subspace; two such spaces are pairwise orthogonal when they are orthogonal in the relevant function-space inner product. The group-action decomposition conjecture. There exists a collection $\mathscr{G}$ of closed, pairwise orthogonal, and minimal spaces of continuous functions on $X$, each invariant under the group action, such that any closed space of continuous functions on $X$ invariant under the group action is the closed direct sum of a unique subcollection of $\mathscr{G}$.