  We consider translation-invariant splitting Gibbs measures (TISGMs) for the
$q$-state Potts model on a Cayley tree of order two. Recently a full
description of the TISGMs was obtained, and it was shown in particular that at
sufficiently low temperatures their number is $2^{q}-1$. In this paper for each
TISGM $\mu$ we explicitly give the set of boundary conditions such that
limiting Gibbs measures with respect to these boundary conditions coincide with
$\mu$.
