The author considers a many-component linear conjugation problem of the form \[ (1)\quad F^+(t)=G(t)F^-(t),\quad | t| =1 \] in the Hölder space \(H_{\nu}({\mathcal L})\), \({\mathcal L}=\{t:| t| =1\}\), \(0<\nu <1\). Here \(G(t)=(G_{ij}(t))_{i,j}\in H_{\nu}(L)\), \(\det G(t)\neq 0\), \(t\in {\mathcal L}\) and the indices of diagonal elements \(G_{ii}(t)\) are positive. He establishes asymptotic properties at infinity for the solution of the problem (1) constructed in the form of a series of piecewise holomorphic functions.