  Let $K$ be a complete discrete valuation field of characteristic $(0, p)$
with perfect residue field, and let $T$ be an integral
$\mathbb{Z}_p$-representation of $\mathrm{Gal}(\overline{K}/K)$. A theorem of
T. Liu says that if $T/p^n T$ is torsion semi-stable (resp. crystalline) of
uniformly bounded Hodge-Tate weights for all $n \geq 1$, then $T$ is also
semi-stable (resp. crystalline). In this note, we show that we can relax the
condition of "uniformly bounded Hodge-Tate weights" to an unbounded
(log-)growth condition.
