Iteration of relative norms through intermediate subgroups

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 whose underlying $K$-equivariant orthogonal spectrum is cofibrant in the $U_K$-universe. Relative norm iteration conjecture. There is a natural $U$-universe $G$-equivariant weak equivalence

$$N_K^G X\simeq N_H^G I_{\mathbb{R}^{\infty}}^{U_H}\bar B\bigl(H/K\times D(T_{eH}(G/H));I_{U_K}^{\mathbb{R}^{\infty}}X\bigr).$$

This asserts that the direct relative norm factors through the intermediate subgroup using factorization homology over $H/K\times D(T_{eH}(G/H))$.