Sharp norm conjecture for probabilistic continuous second-order Riesz transforms

Let $d\ge 2$, $1<p<\infty$, and let $\mathbf{R}^{(jk)}$ be the probabilistic continuous second-order Riesz transforms on $\mathbb{R}^d$. Let $p^*=\max\\{p,p/(p-1)\\}$, and let $\gamma(p)$ be Choi's sharp constant for non-symmetric martingale transforms with predictable multipliers in $[0,1]$.

Probabilistic continuous Riesz-transform conjecture. For $j\ne k$,\

$$\\|2\mathbf{R}^{(jk)}\\|_{L^p\to L^p}=p^*-1,$$

\

$$\\|\mathbf{R}^{(jj)}-\mathbf{R}^{(kk)}\\|_{L^p\to L^p}=p^*-1.$$

For diagonal transforms,\

\\|\mathbf{R}^{(jj)}\\|_{L^p\to L^p}=\gamma(p).\

These would extend the sharp classical second-order Riesz-transform norm identities to the probabilistic continuous operators. The paper currently gives upper bounds differing by dimensional constants, so the conjectured sharp equalities remain open.