Peter Scholze formulated several conjectures about $q$-de Rham complex in the paper
Especially intriguing to me is Conjecture 3.3: Betti-de Rham comparison isomorphism.
In the paper the author emphasized the lack of nontrivial examples and suggests to do a computation for the universal family of elliptic curves.
Question 1: Has anyone done this computation (or any other interesting computations)? I will be especially interested in those involving basic hypergeometric functions.
Question 2: What does Conjecture 3.3 mean in practice? Naively one might hope that it leads to some $q$-deformation of periods.