For $K$ a $\mathfrak p$-adic number field, local Tate duality yields a non-degenerate pairing
$$H^1(K, \Bbb Z/n\Bbb Z) \times H^1(K, \mu_n) \longrightarrow \Bbb Z/n\Bbb Z,$$
where $\mu_n$ is the group of $n$-th roots of unity, given by $$(\chi, a) \mapsto \chi((a, \bar K|K)).$$
Here $H^1(K, A)$ stands for $H^1(\operatorname{Gal}(\bar K|K), A)$, $\chi$ is a character $\chi: \operatorname{Gal}(\bar K|K) \longrightarrow \Bbb Z/n\Bbb Z$, $a$ lies in $K^\times / K^{\times n}$, and $(a, \bar K|K)$ is the local Artin symbol.
If $n$ doesn't divide the characteristic of the residue field of $K$ then the orthogonal complement of the unramified cohomology $H^1_{nr}(K, \Bbb Z/n\Bbb Z) = H^1(\operatorname{Gal}(\tilde K|K), \Bbb Z/n\Bbb Z)$ is $H^1_{nr}(K, \mu_n) = H^1(\operatorname{Gal}(\tilde K|K), \mu_n).$
Now for a number field $K$, this gives us a non-degenerate pairing of locally compact groups $$\prod'_{\mathfrak{p}} H^1(K_{\mathfrak{p}}, \mathbb{Z}/n\mathbb{Z}) \times \prod'_{\mathfrak{p}} H^1(K_{\mathfrak{p}}, \mu_n) \rightarrow \mathbb{Z}/n\mathbb{Z}$$
given by $$(\chi, \alpha) = \sum_{\mathfrak{p}} \chi_{\mathfrak{p}}(\alpha_{\mathfrak{p}}, \bar{K_{\mathfrak{p}}}/K_{\mathfrak{p}}),$$
where the restricted products are taken with respect to the unramified cohomology groups defined above.
It turns out that the images of $$H^1(K, \mathbb{Z}/n\mathbb{Z}) \rightarrow \prod'_{\mathfrak{p}} H^1(K_{\mathfrak{p}}, \mathbb{Z}/n\mathbb{Z})$$
and
$$H^1(K, \mu_n) \rightarrow \prod'_{\mathfrak{p}} H^1(K_{\mathfrak{p}}, \mu_n)$$
are mutual orthogonal complements with respect to the global pairing.
So far, so good. Now given the above, Neukirch set the following exercise on p. 404 of Algebraic Number Theory:
If $S$ is a finite set of places of $K$, then the map
$$H^1(K, \mathbb{Z}/n\mathbb{Z}) \rightarrow \prod'_{\mathfrak{p} \in S} H^1(K_{\mathfrak{p}}, \mathbb{Z}/n\mathbb{Z})$$
is surjective if and only if the map
$$H^1(K, \mu_n) \rightarrow \prod'_{\mathfrak{p} \notin S} H^1(K_{\mathfrak{p}}, \mu_n)$$
is injective.
I know this can be proved using Tate-Poitou Duality as in Neukirch-Schmidt-Wingberg, but I have made no progress trying to prove this given only the facts outlined above. I would appreciate any and every hint.