$L^p$ boundedness conjecture for geometric maximal operators of curve averages

Let $n\geq 2$ and let $\gamma:I \to \mathbb R^n$ be a non-degenerate curve. Define the geometric maximal operator

$$M_\gamma f(x):=\sup_{t>0}|A_t f(x)|.$$

Maximal-operator conjecture. The operator $M_\gamma$ maps $L^p(\mathbb R^n)$ to $L^p(\mathbb R^n)$ boundedly if and only if $p>n$. This conjecture concerns the $L^p$ theory of geometric maximal operators and generalizes Bourgain's circular maximal function to higher dimensions. The source does not state a resolution or partial range for this conjecture.