Geometric fixed points of relative norms for normal subgroups

Let $G$ be a compact Lie group, let $H<G$ be a closed subgroup, let $K\trianglelefteq G$ be normal, and let $X$ be an $H$-equivariant $\mathcal{D}_V$-algebra satisfying the paper's Cofibrancy Hypothesis. If $H$ has finite index in $HK$, then $H\cap K$ acts trivially on $V$, the map of $H$-equivariant inner product spaces

$$V=T_{eH}G/H\longrightarrow T_{eHK}G/HK$$

is an isomorphism, and there is a natural $U^K$-universe $G/K$-equivariant weak equivalence

$$(N_H^G X)^{\Phi K}\simeq N_{(G/K)/(HK/K)}^{G/K}(X^{\Phi(H\cap K)}).$$

If $H$ is not finite index in $HK$, the unit map instead induces a $U^K$-universe $G/K$-equivariant weak equivalence $\mathbb{S}\longrightarrow (N_H^G X)^{\Phi K}$. This conjecture describes geometric fixed points of relative norms in the normal-subgroup cases.