Galois invariance of the Det-part of K₁-groups of Iwasawa algebras

Let $L$ be either an arbitrary Galois extension $L^0$ of finite absolute ramification index over $\mathbb{Q}_p$ or the completion of such an extension, let $S=\mathcal{O}_L$, and let $G$ be a finite group. Let $\Delta$ be an open subgroup of $\operatorname{Gal}(L^0/\mathbb{Q}_p)$ acting coefficientwise on $S[G]$, and hence on the associated Det-groups. Galois invariance conjecture. The inclusion $i:S^\Delta[G]\hookrightarrow S[G]$ induces an isomorphism

$$i_*:\operatorname{Det}(S^\Delta[G]^\times)\cong\operatorname{Det}(S[G]^\times)^\Delta.$$

This predicts that the Det-part of the $K_1$-group over the fixed coefficient ring is exactly the subgroup of Galois invariants over $S$. The parser supplies no evidence resolving the assertion, so its status remains open.