De Rham-Higgs comparison for mixed Hodge modules in positive characteristic

Building on the nonabelian Hodge theory in positive characteristic developed by Ogus, Vologodsky, and Schepler, we propose a generalization of the decomposition theorem of Deligne and Illusie from the perspective of mixed Hodge modules. This generalization is verified in certain special cases by extending the method of Sheng and the author, leading to several interesting byproducts.