Conjecture on the $L^p$-boundedness of the generalized spherical maximal operator

Let $n\ge 2$ and $1< p < \infty$. Let $M^{\beta}_*$ denote the generalized spherical maximal operator on $\mathbb{R}^n$, and let condition (cnd18) be

$$p \le \frac{2n-1}{n-1} \quad\text{and}\quad \beta > 1-n+n/p, \quad\text{or}\quad p > \frac{2n-1}{n-1} \quad\text{and}\quad \beta > \frac{1-n}{p},$$

with the last inequality weakened when $-\beta \in \mathbb{N}$ or $\beta > \frac{3-n}{2}$. Generalized spherical maximal conjecture. The operator $M^{\beta}_*$ is bounded on $L^p(\mathbb{R}^n)$ if and only if condition (cnd18) is satisfied. This conjecture asks whether the sharp radial-function bounds extend to all functions; known sharp results for $M^0_*$ motivate the claim, while the equivalence for general $\beta$ remains open.