Uniform restricted X-ray transform estimates for polynomial curves

Let $d\geq 3$ and let $P:{\bf R}\to{\bf R}^{d-1}$ be a polynomial of degree $N$. Write

$$d\gamma^*\lambda(t)=|\det(\gamma'(t),\ldots,\gamma^{(d-1)}(t))|^{\frac{2}{d(d-1)}}\,dt$$

for affine arclength measure, and let $X^P$ denote the restricted X-ray transform associated with $P$. Let $p,q,r$ satisfy the conditions in the source, with equality in each of the first two conditions and also the third condition. Uniform restricted X-ray transform conjecture. Then

$$\|X^P f\|_{L^q(L^r;d\gamma^*\lambda)}\leq C\|f\|_{L^p}$$

for every $f\in L^p$, where $C$ depends only on $d$, $N$, and $\theta$. Furthermore, if $P$ is fixed and $L_P$ is not identically zero, these are the only exponents for which the asserted estimate can hold. The conjecture proposes affine-arclength-weighted mixed-norm bounds uniform over polynomial curves of fixed degree; the supplied text does not establish the estimate or provide evidence resolving the conjecture.