Theory of weights for log convergent cohomologies II: the case of a proper SNCL scheme in characteristic $p>0$

For a flat $p$-adic formal family $S$ of log points over a complete discrete valuation ring with perfect residue field of mixed characteristics $(0,p)$ and for a simple normal crossing log scheme $X$ over an exact closed log subscheme of $S$ defined by an element of the maximal ideal of the dvr, we construct two fundamental filtered complexes in the convergent topos the underlying scheme of $X$ over the underlying scheme of $S$. We prove that they are canonically isomorphisc. Because one of the complex is shown to calculate the log convergent cohomology sheaf of $X\mathrm{/}S$, the filtered complex produces the weight filtration on the log convergent cohomology sheaf if the underlying scheme of $X$ is proper over the underlying scheme of $S$. We give a comparison theorem between the projection of the filtered complex in the Zariski topos of the underlying scheme of $X$ and the isozariskian weight-filtered complex constructed in the author’s previous work.