Base change of twisted Fontaine-Faltings modules and Twisted Higgs-de Rham flows over very ramified valuation rings

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)}$,eachstabletwisted$f$-periodic logarithmic Higgs bundle $(E,\theta )$ over the closed fiber $(X \supset D){$\mathbb{F}$q}$willcorrespondtoa$$\mathrm{P}\mathrm{G}\mathrm{L}$r($\mathbb{F}${p^f})$-crystallinerepresentationof$\pi_1(($\mathcal{X}$ \setminus $\mathcal{D}$){W($\mathbb{F}$_q)[\frac{1}{p}]})$ such that its restriction to the geometric fundamental group is absolutely irreducible.