Geometricity of the Hodge filtration on the $\infty$-stack of perfect complexes over $X_\mathrm{DR}$

We construct a locally geometric $\infty$-stack $\mathscr M_\mathrm{Hod}(X,\mathrm{Perf})$ of perfect complexes with $\lambda$-connection structure on a smooth projective variety $X$. This maps to $\mathbb A^1/\mathbb G_m$, so it can be considered as the Hodge filtration of its fiber over 1 which is $\mathscr M_\mathrm{DR}(X,\mathrm{Perf})$, parametrizing complexes of $\mathscr D_X$-modules which are $\mathscr O_X$-perfect. We apply the result of Toen–Vaquié that $\mathrm{Perf}(X)$ is locally geometric. The proof of geometricity of the map $\mathscr M_\mathrm{Hod}(X,\mathrm{Perf})\to\mathrm{Perf}(X)$ uses a Hochschild-like notion of weak complexes of modules over a sheaf of rings of differential operators. We prove a strictification result for these weak complexes, and also a strictification result for complexes of sheaves of $\mathscr O$-modules over the big crystalline site.