The sharp norm conjecture for the $(j,k)$-plane transform

Let $0\le j<k<n$. For a measurable function $f$ on the affine Grassmannian $\mathcal A_{n,j}$, let $R_{j,k}f$ denote its $(j,k)$-plane transform on $\mathcal A_{n,k}$, and set

$$p=\frac{n+1}{k+1},\qquad q=\frac{n+1}{j+1},\qquad \Omega_{j,k}(n)=\left(\sigma_j^{k-n}\sigma_k^{n-j}\sigma_n^{j-k}\right)^{1/(n+1)}.$$

The sharp norm conjecture for the $(j,k)$-plane transform. One has

$$\lVert R_{j,k}f\rVert_{L^q(\mathcal A_{n,k})}\leq \Omega_{j,k}(n)\,\lVert f\rVert_{L^p(\mathcal A_{n,j})},$$

where $\Omega_{j,k}(n)$ is the norm of $R_{j,k}$, and equality holds if and only if

$$f(\zeta)=c\,(1+\lvert M\zeta\rvert^2)^{-(k+1)/2},\qquad \zeta\in\mathcal A_{n,j},$$

with $c$ constant and $M$ an invertible affine map.

This conjecture would give a complete analogue for $(j,k)$-transforms of the sharp $k$-plane transform theorem, including both the optimal constant and all extremizers. The source states that this analogue is unknown; no resolution is supplied here.