Coordinate independence of the q-de Rham complex

Let $R$ be a smooth $\mathbb Z$-algebra, and let $\square:\mathbb Z[T_1,\ldots,T_d]\to R$ be an étale framing. The framed complex $q\!\operatorname-\!\Omega^\bullet_{R[[q-1]]/\mathbb Z[[q-1]],\square}$ is a complex of $\mathbb Z[[q-1]]$-modules. The coordinate-independence conjecture. The framed complex is independent of the choice of coordinates up to canonical quasi-isomorphism; more precisely, there is a functor $R\mapsto q\!\operatorname-\!\Omega_R$ to the $\infty$-category of $E_\infty$-$\mathbb Z[[q-1]]$-algebras, computed by the framed complex for every framing. This would make the construction intrinsic and allow it to glue over smooth schemes.