Smooth centralizer property conjecture for finite orthogonal groups

Let $G$ be a finite subgroup of $\operatorname{O}(n)$. Say that $G$ has the amphichiral linear centralizer property (LCP) if its centralizer in $\operatorname{O}(n)$ contains an element outside $\operatorname{SO}(n)$, and say that $G$ has the amphichiral smooth centralizer property (SCP) if its centralizer in $\operatorname{Diff}(S^{n-1})$ contains an orientation-reversing diffeomorphism. Smooth centralizer property conjecture. If $G$ has SCP, then it has LCP. Equivalently, if the centralizer of $G$ in $\operatorname{Diff}(S^{n-1})$ is not contained in $\operatorname{Diff}^{+}(S^{n-1})$, then it contains an element of $\operatorname{O}(n)\setminus\operatorname{SO}(n)$. This is an equivalent centralizer formulation relevant to the main conjecture. Since the source gives no resolution beyond the stated low-dimensional result, the general assertion remains open.