Generalized-polynomial extension of the integer-part polynomial convergence theorems

Let the systems, functions, and generalized-polynomial iterates $p_{i,j}$ be as in Theorems 1* and 3* of the source paper, which concern mean convergence for multiple ergodic averages with integer-part polynomial iterates.

Generalized-polynomial convergence conjecture. Theorems 1* and 3* hold if $p_{i,j}$ are generalized polynomials.

This conjecture proposes extending the paper's convergence results from polynomial iterates to generalized-polynomial iterates. The source presents it as a conjecture and gives no resolution.