Hardy 2-corners: identity and norm convergence remain unknown

Establish whether for the Hardy sequence a(n)=⌊n^{3/2}⌋ and any system (X,𝔛,μ,T_1,T_2) and functions f_1,f_2∈L^∞(μ) the identity lim_{N→∞} (1/N)∑_{n=1}^N T_1^{a(n)}f_1·T_2^{a(n)}f_2 − (1/N)∑_{n=1}^N T_1^n f_1·T_2^n f_2 = 0 holds in L^2(μ), and determine whether the left-hand average converges in L^2(μ).

Background

For corners (simultaneous iterates under several transformations with the same parameter), comparison identities analogous to those for arithmetic progressions are highly nontrivial.

Even in the basic Hardy case a(n)=n^{3/2}, neither the identity nor L^2 convergence is presently known.