Relative norm invariance under cofibrant weak equivalences

Let $G$ be a compact Lie group, let $H<G$ be a closed subgroup, and let $N_H^G$ be the point-set relative norm indexed on universes $U_H$ and $U$. Suppose $X'$ and $X$ satisfy the paper's Cofibrancy Hypothesis. Relative norm homotopy-invariance conjecture. A $U_H$-universe weak equivalence $X'\longrightarrow X$ induces a $U$-universe weak equivalence

$$N_H^G X'\longrightarrow N_H^G X.$$

This would establish that the point-set relative norm has the expected homotopy invariance on the specified cofibrant inputs and hence supports a derived functor.