A stacky comparison of the Hodge and Nygaard filtrations

We use the approach to $p$-adic cohomology theories via stacks recently developed by Drinfeld and Bhatt–Lurie to formulate a stacky version of a comparison result between the Nygaard filtration on prismatic cohomology and the Hodge filtration on de Rham cohomology by Bhatt–Lurie and thereby also obtain a generalisation in the case of smooth and proper $p$-adic formal schemes which allows for coefficients in an arbitrary gauge. In the process, we develop a stacky approach to diffracted Hodge cohomology as introduced by Bhatt–Lurie which also captures the conjugate filtration and the Sen operator. In the appendix, we also introduce a stack computing the conjugate filtration on absolute Hodge–Tate cohomology.