Pointwise convergence and maximal cone multiplier conjecture

Let $n\geq3$, $p\geq2$, and define

$$\widetilde T_t^\lambda(f)(x)=\int_{\mathbb{R}^n}\widetilde m^\lambda(t^{-1}\xi',\xi_n)\widehat f(\xi)e^{2\pi i x\cdot\xi}\,d\xi,$$

where $\widetilde m^\lambda$ is the cone multiplier. Also define

$$\widetilde T_*^\lambda(f)(x)=\sup_{t>0}|\widetilde T_t^\lambda(f)(x)|.$$

Pointwise convergence and maximal cone multiplier conjecture. The range of $p$ for which

$$\lim_{t\to\infty}\widetilde T_t^\lambda(f)(x)=f(x)\qquad\text{a.e.}$$

for every $f\in L^p(\mathbb{R}^n)$ is the same as the range for which $\widetilde T^\lambda$ is bounded on $L^p(\mathbb{R}^n)$, namely

$$\lambda>\max\left\{(n-1)\left|\frac1p-\frac12\right|-\frac12,0\right\}.$$

Moreover, $\widetilde T_*^\lambda$ is bounded on $L^p(\mathbb{R}^n)$ whenever the same inequality holds. This conjecture asks for the sharp pointwise and maximal estimates associated with cone multipliers; the supplied text does not state that it has been resolved.