The degree-uniform improving conjecture for polynomial averages

Let $d\geq 2$ be an integer, let $p(x)$ be any polynomial of degree $d$ mapping the integers to the integers, and let $N\geq 1$. Define

$$A_N f(x)=\frac{1}{N}\sum_{n=1}^{N}f(x+p(n)).$$

For the interval $I=[0,N^d]$, let $f$ be supported on $I$, and write $q'$ for the Hölder conjugate of $q$. The degree-uniform improving conjecture. There exists an exponent $q=q_d$ with $1<q<2$, depending only on $d$, such that, uniformly in $N$ and in the polynomial $p$,

$$\langle A_Nf\rangle_{I,q'}\lesssim\langle f\rangle_{I,q}.$$

This conjecture asks for an $\ell^p$-improving estimate for averages along arbitrary integer-valued polynomial sequences, with the improving exponent controlled only by the polynomial degree. The source does not provide a resolution, so the conjecture remains open.