Strong-type endpoint conjecture for finite-field $k$-plane transforms

Let $F_q$ be a finite field, let $dgeq 2$ and $1\leq kleq d-1$ be integers, and let $T_{Pi_k}$ denote the $k$-plane transform on the collection $Pi_k$ of all $k$-planes in $F_q^d$, with the normalized measures $dx$ and $dsigma$. The conjecture asserts that, for every function $f$ on $F_q^d$,

$$\|T_{\Pi_k}f\|_{L^{d+1}(\Pi_k,d\sigma)}\leq C\|f\|_{L^{\frac{d+1}{k+1}}(\mathbb F_q^d,dx)},$$

where $C$ is independent of $|\mathbb F_q|$. This would upgrade the known restricted-type endpoint estimate to a strong-type estimate and complete the mapping properties of finite-field $k$-plane transforms at the endpoint.