  In this short notes, we prove a stronger version of Theorem 0.6 in our
previous paper arXiv:1709.01485: Given a smooth log scheme $(\mathcal{X}
\supset \mathcal{D})_{W(\mathbb{F}_q)}$, each stable twisted $f$-periodic
logarithmic Higgs bundle $(E,\theta)$ over the closed fiber $(X \supset
D)_{\mathbb{F}_q}$ will correspond to a
$\mathrm{PGL}_r(\mathbb{F}_{p^f})$-crystalline representation of
$\pi_1((\mathcal{X} \setminus \mathcal{D})_{W(\mathbb{F}_q)[\frac{1}{p}]})$
such that its restriction to the geometric fundamental group is absolutely
irreducible.
