The $n/k$ to $n$ norm conjecture for the maximal operator ${\mathcal N}^k$

Let ${\mathbb F}$ be ${\mathbb Z}_p$ or $\hat{{\mathbb Z}}$, let $k\ge2$, and let ${\mathcal N}^k$ be the maximal operator over $k$-dimensional subspaces of ${\mathbb F}^n$. For a function $f:{\mathbb F}^n\to{\mathbb C}$, write $\operatorname{Gr}({\mathbb F}^n,k)$ for the corresponding Grassmannian, and let $\mu$ and $\nu$ be normalized Haar measures on ${\mathbb F}^n$ and $\operatorname{Gr}({\mathbb F}^n,k)$.

The $n/k$ to $n$ norm conjecture for ${\mathcal N}^k$. There is a constant $C_n$, depending on $n$ and ${\mathbb F}$, such that

$$C_n\int\limits_{U\in \operatorname{Gr}({\mathbb F}^n,k)} |{\mathcal N}^k f(U)|^{n}\,d\nu\le \left(\int\limits_{x\in {\mathbb F}^n}|f(x)|^{n/k}\,d\mu\right)^k.$$

The paper proves an $n-1$ to $n-1$ estimate for $k=2$, but explicitly states that it does not obtain bounds strong enough to prove this conjecture.