Functoriality of the q-Hodge complex without compatibility with q–de Rham–Witt structure

Determine whether the q-Hodge complex obtained by multiplying the differentials in the de Rham complex by (q−1) can be defined functorially (as an object in the appropriate derived category) for framed smooth Z-algebras without requiring compatibility with the extra q–de Rham–Witt structure established in Theorem 1.3.

Background

In Section 1.1 the authors explain a no-go theorem showing that the q-Hodge complex cannot be made functorial in a way that preserves a wealth of additional q–de Rham–Witt structures. Nevertheless, they note a residual possibility: functoriality might still be possible if one drops compatibility with that extra structure.

Clarifying whether any functorial construction of the q-Hodge complex exists (even if incompatible with the q–de Rham–Witt package) would settle the extent of the obstruction identified by the no-go theorem.