Let $K$ be a non-archimedean local field of characteristic 0. Fix a uniformiser $\pi$ and an algebraic closure $\bar{K}$. The theory of Lubin--Tate formal groups gives an explicit construction of the tower $K\subset K_{\pi,1}\subset K_{\pi,2}\subset\cdots$ so that the maximal abelian extension $K^{ab}$ is the same as the composite $K_\pi\cdot K^{ur}$ where $K_\pi=\bigcup K_{\pi,n}$ and $K^{ur}$ is the maximal unramified extension.
As you may know there is a construction of the local reciprocity $K^\times\to \mathrm{Gal}(K^{ab}/K)$ using group cohomology.
I am wondering if there exists a way to recover the tower $\{K_{\pi,n}\}$ using the group cohomology.
I am pretty sure that this question is natural enough so that it should be asked to the community after the local class field theory was established. Any reference would be greatly helfpul.
