Conjecture on Lévy-spherical multipliers

Let $\mathbb{S}$ be the unit sphere in $\mathbb{R}^n$, let $\varphi:\mathbb{S}\to\mathbb{C}$ satisfy $\|\varphi\|_{L^{\infty}}\leq 1$, let $0<r<\infty$ and $n\geq 2$, and let $\sigma$ be surface measure on $\mathbb{S}$. Define

$$\mathcal{M}(\xi)=\frac{\int_{\mathbb{S}}|\xi\cdot\theta|^r\varphi(\theta)\,d\sigma(\theta)}{\int_{\mathbb{S}}|\xi\cdot\theta|^r\,d\sigma(\theta)}.$$

Let $\mathcal{S}_{\mathcal{M}}$ denote the Fourier multiplier operator with multiplier $\mathcal{M}$. Lévy-spherical multiplier conjecture.

$$\|\mathcal{S}_{\mathcal{M}}\|_p\leq p^*-1.$$

The conjecture is proposed because it would imply the desired sharp bound for the Beurling–Ahlfors operator. The source supplies no resolution.