Hochman's upcrossing theorem for groups of polynomial growth

Consider a stochastic process $(S_{[a_i,b_i]})_{[a_i,b_i] \subset \mathbb{N}}$, which is indexed by the collection of all nonempty intervals $[a_i,b_i] \subset \mathbb{N}$ and which is stationary under translations of the intervals. It was shown by M. Hochman that, for any $k \geq 1$ and any interval $(\alpha,\beta) \subset \mathbb{R}$, one can give an `almost-exponential' bound on the size of the set where the associated process $(S_{[1,n]})_{n \geq 1}$ has at least $k$ fluctuations over $(\alpha,\beta)$. It was also noticed that a similar techniques can be applied in $\mathbb{Z}^d$ case. In this article we extend Hochman's upcrossing theorem to groups of polynomial growth.