Pointwise kernel domination conjecture for probabilistic discrete Riesz transforms

For $d\geq2$, let $K_{\mathbb{H}^{(k)}}$ and $K_{R_\mathrm{dis}^{(k)}}$ be the kernels of the probabilistic and Calderón–Zygmund discrete Riesz transforms, respectively, with $n\in\mathbb{Z}^d$ and $k=1,\dots,d$. Pointwise kernel domination conjecture. For every $n\in\mathbb{Z}^d$,

$$\left|K_{\mathbb{H}^{(k)}}(n)\right|\geq\left|K_{R_\mathrm{dis}^{(k)}}(n)\right|.$$

The comparison is supported by numerical simulations and is known when $d=1$; the conjecture concerns all dimensions $d\geq2$.