Sobolev improving conjecture for non-degenerate curves

Let $n\geq 2$, let $I\subset\mathbb R$ be a compact interval, let $\gamma\colon I\to\mathbb R^n$ be a smooth curve, and let $\chi\in C^{\infty}(\mathbb R)$ be a bump function supported on the interior of $I$. Define

$$A_{\gamma}f(x):=\int_{\mathbb R}f(x-\gamma(s))\chi(s)\,\mathrm{d}s.$$

The curve $\gamma$ is non-degenerate if there is a constant $c_0>0$ such that

$$|\det(\gamma'(s),\ldots,\gamma^{(n)}(s))|\geq c_0\qquad\text{for all }s\in I.$$

Non-degenerate curve Sobolev improving conjecture. If $\gamma$ is non-degenerate and $2n-2<p<\infty$, then

$$\|A_{\gamma}f\|_{L^p_{1/p}(\mathbb R^n)}\lesssim_{p,\gamma,\chi}\|f\|_{L^p(\mathbb R^n)}.$$

This would extend the theorem proved in the paper for curves in $\mathbb R^4$, where the estimate holds for $6<p<\infty$. The conjecture concerns the currently unknown higher-dimensional range of sharp $L^p$-Sobolev improving estimates for averaging over non-degenerate curves.