Universal Gaussian off-diagonal decay conjecture for Toeplitz operators

Let $d$ and $N$ be integers, and let $U,V$ be open subsets of $(\mathbb{S}^2)^d$. Write $\operatorname{dist}(U,V)$ for their distance and let $S_N$ denote the Toeplitz operator appearing above. Universal Gaussian decay conjecture. There exists a universal constant $c>0$ such that

$$\|\mathds{1}_U S_N\mathds{1}_V\|_{L^2\to L^2}\leq \exp\bigl(-cN\operatorname{dist}(U,V)^2\bigr).$$

This would strengthen the preceding dimension-dependent estimate by giving Gaussian off-diagonal decay with a constant independent of $d$ and $N$. The source presents it as a conjecture after noting that the Schur-test estimate is crude; no resolution is supplied here.