Conjecture on regularity over the cyclic group of order a power of two

Let $f\colon \mathbb{Z}/2^n\mathbb{Z}\to\{\pm1\}$ be a Boolean function. Say that $f$ has $(\delta,k)$-$\mathbb{R}$-regularity and that it has $\epsilon$-$\mathbb{Z}/2^n\mathbb{Z}$-Regularity according to the regularity notions defined in the paper. Regularity transfer conjecture. For any $\epsilon>0$, there is a $k=k(\epsilon)$ and a $\delta=\delta(\epsilon)>0$ such that $(\delta,k)$-$\mathbb{R}$-regularity implies $\epsilon$-$\mathbb{Z}/2^n\mathbb{Z}$-Regularity. The conjecture is motivated largely by numerical evidence and asks for an analogue over $\mathbb{Z}/2^n\mathbb{Z}$ of the quasirandomness results established over $\mathbb{F}_2^n$; its general validity remains open.