Factorization-homology iteration for equivariant norms

Let $G$ be a compact Lie group, let $K<H<G$, and let $X$ be a $K$-equivariant little $T_{eK}(G/K)$-disk algebra. Let $D(T_{eH}(G/H))$ be the open unit disk and let $i^*$ denote the functor modeling the homotopical forgetful functor associated to the tangent-space decomposition. Factorization-homology iteration conjecture. There is a natural zigzag of $H$-equivariant homotopy equivalences

$$\bar B(H/K\times D(T_{eH}(G/H));X)\simeq B(H/K,i^*X).$$

This comparison is intended to resolve the mismatch between the disk-algebra structures needed for $N_K^G$ and $N_K^H$, enabling an iterated description of relative norms.