The invariant little Grothendieck conjecture for invariant function spaces

Let $G$ be a compact abelian group, let $Z(G)$ be an invariant function space, and let $T:H_1\to H_2$ be an operator between Hilbert spaces. Say that $T$ factorizes invariantly through $Z(G)$ if there are orthonormal bases $(h_{n,1})$ and $(h_{n,2})$ of $H_1$ and $H_2$, operators $A:H_1\to Z(G)$ and $B:Z(G)\to H_2$ with $T=B\circ A$, and an enumeration $(\gamma_n)$ of the characters of $G$ such that $A(h_{n,1})\in\operatorname{span}\gamma_n$ and $B(\gamma_n)\in\operatorname{span}h_{n,2}$ for $n=1,2,\dots$. Invariant little Grothendieck conjecture. If $Z(G)$ has no complemented, invariant infinite-dimensional subspaces isomorphic to a Hilbert space, then every Hilbert space operator $T$ that factorizes invariantly through $Z(G)$ belongs to some nontrivial Schatten class. The claim abstracts the Sobolev-space question to invariant function spaces; the paper presents it as the conjectural interpretation motivating its harmonic-analysis result, while the invariant little Grothendieck theorem for $L_1$ is noted as known.