Subcritical p-improving estimate conjecture for discrete polynomial curves

Let $d\geq 1$ and let $\gamma(n)=(P_1(n),\ldots,P_d(n))$ be a polynomial curve in $\mathbb{Z}^d$, where $P_1,\ldots,P_d\in\mathbb{Z}[X]$ have separated degrees, meaning $\deg P_j<\deg P_{j+1}$ for $1\leq j<d$. Define the discrete average

$$\mathcal{A}^{\gamma}_{N}f(\bm{x})=\frac{1}{N}\sum_{n=1}^{N}f(\bm{x}-\gamma(n)).$$

Writing $D=D_{\gamma}:=\sum_{j=1}^{d}\operatorname{deg}P_j$ for the total degree of $\gamma$, and letting $q'$ satisfy $1/q+1/q'=1$, the subcritical $\ell^p$-improving conjecture. For any exponents $p,q$ with $q\geq p$ and every $\epsilon>0$,

$$\|\mathcal{A}_Nf\|_{\ell^q(\mathbb{Z}^d)}\lesssim_{\epsilon}N^{\epsilon}\left(N^{-D(1/p-1/q)}+N^{-1/q'}+N^{-1/p}\right)\|f\|_{\ell^p(\mathbb{Z}^d)}$$

holds for every $f\in\ell^p(\mathbb{Z}^d)$. The three powers of $N$ are forced by testing on a Dirac delta, on the characteristic function of the image $\gamma([1,N])$, and on a suitable dilate of a parabolic box, respectively; the conjecture asserts that these necessary powers are sufficient up to an $N^{\epsilon}$ loss.