The prismatic–de Rham–Witt comparison conjecture for special perfect prisms

Let $(A,d)$ be a perfect prism such that $A/d$ is $p$-torsion free, and let $S$ be a smooth $A/d$-algebra. Here $W_n\Omega_{S/(A/d)}^i$ denotes the degree-$i$ de Rham–Witt group, and $H^i({\mathlarger{\mathbbl{\Delta}}}_{S/A}\otimes^L_A A/d\ldots\phi^{n-1}(d))$ denotes the degree-$i$ cohomology of the indicated prismatic complex after derived base change. Prismatic–de Rham–Witt comparison conjecture. There is a functorial isomorphism

$$W_n\Omega_{S/(A/d)}^i\to H^i({\mathlarger{\mathbbl{\Delta}}}_{S/A}\otimes^L_A A/d\ldots\phi^{n-1}(d)).$$

This conjecture proposes a comparison between de Rham–Witt forms and prismatic cohomology for smooth algebras over the quotient of a perfect prism. Its status is not resolved by the supplied source context.