  Let $K$ be a field with $G_K(2) \simeq G_{\mathbb{Q}}(2)$, where $G_F(2)$
denotes the maximal pro-2 quotient of the absolute Galois group of a field $F$.
We prove that then $K$ admits a (non-trivial) valuation $v$ which is
2-henselian and has residue field $\mathbb{F}_2$. Furthermore, $v(2)$ is a
minimal positive element in the value group $\Gamma_v$ and
$[\Gamma_v:2\Gamma_v]=2$. This forms the first positive result on a more
general conjecture about recovering $p$-adic valuations from pro-$p$ Galois
groups which we formulate precisely. As an application, we show how this result
can be used to easily obtain number-theoretic information, by giving an
independent proof of a strong version of the birational section conjecture for
smooth, complete curves $X$ over $\mathbb{Q}_2$, as well as an analogue for
varieties.
