  Let $p>2$ be a prime. Let $\rho$ be a crystalline representation of
$G_{\mathbb{Q}_p}$ with distinct Hodge-Tate weights in $[0, p]$, such that its
reduction $\overline \rho$ is upper triangular. Under certain conditions, we
prove that $\overline \rho$ has an upper triangular crystalline lift $\rho'$
such that $\mathrm{HT}(\rho')=\mathrm{HT}(\rho)$. The method is based on the
author's previous work, combined with an inspiration from the work of
Breuil-Herzig.
