Let $M/L/\mathbb{Q}_p$ be a finite galois abelian extension of local fields and define
$\mathcal{M}=M\{\{T\}\}=\{\sum_{i\in \mathcal{Z}}a_iT^i : a_i\in M, \min_{i\in \mathcal{Z}} v(a_i)>-\infty, \lim_{i\to -\infty} v(a_i)=\infty\}$.
Then $\mathcal{M}/\mathcal{L}$, $\mathcal{L}=L\{\{T\}\}$, is also a finite abelian galois extension. Then Kato's reciprocity map induces an isomorphism
$\Upsilon_{\mathcal{L}}:K_2(\mathcal{L})/N(K_2(\mathcal{M})) \to Gal(\mathcal{M}/\mathcal{L})$,
where $K_2(\mathcal{L})$ is the $2^{nd} $Milnor K-group of $\mathcal{L}$ and $N:K_2(\mathcal{M})\to K_2(\mathcal{L})$ is the norm on $K$-Milnor groups.
Since clearly
\begin{align*} Gal(M/L)&\cong Gal(\mathcal{M}/\mathcal{L})\\ g&\to \ g(\sum a_iT^i) =\sum g(a_i)T^i \end{align*}
then my question is, what it is the relation between the 2-dimensional reciprocity map $\Upsilon_\mathcal{L}$ and the 1-dimensional reciprocity map $\theta_{L}:L^*\to Gal(M/L)$ in this scenario? I believe that a relation like
$\Upsilon_{\mathcal{L}} (\{a,T\})=\theta_L(a)$
should be true. I would like to know any reference or suggestion about this.
