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.
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