Joint ergodicity for generalized Hardy sequences on totally ergodic systems (Tsinas Conjecture 2)

Show that for any totally ergodic system (X,𝔛,μ,T) and any 0<b_1<⋯<b_ℓ, the sequences ⌊n^{b_1}⌋^2,...,⌊n^{b_ℓ}⌋^2 are jointly ergodic, i.e., the corresponding multiple averages converge in L^2(μ) to the product of integrals.

Background

Beyond convergence, this conjecture predicts the product-structure limit (joint ergodicity) for averages with generalized Hardy-square iterates on totally ergodic systems.

It parallels established results for polynomial and standard Hardy iterates.