Logarithmic exact crystalline Poincaré lemma of higher level modulo p

Le Stum and Quirós proved the formal Poincaré lemma in crystalline cohomology of higher level by using the jet complex, and this result gives us a de Rham interpretation of this cohomology. In this article, we prove the logarithmic version of the formal Poincaré lemma modulo p. Provided that each term of the log. jet complex is locally free, it gives the logarithmic version of the de Rham interpretation of the crystalline cohomology of higher level.