Exponential bounds conjecture for the tempered fractional kernel equivalence factor

Let $n$ and $s$ be the dimension and fractional-order parameters, let $\boldsymbol{x},\boldsymbol{y}$ be points in the underlying Euclidean space, and let $\lambda$ be the tempering parameter. The function $F(n,s,\lambda,|\boldsymbol{x}-\boldsymbol{y}|)$ is the factor appearing in the equivalence kernel for the tempered fractional operators. Exponential bounds conjecture. There are positive constants $\underline{B}_{n,s}$ and $\overline{B}_{n,s}$ such that

$$\underline{B}_{n,s}e^{-\lambda|\boldsymbol{x}-\boldsymbol{y}|}\leq F(n,s,\lambda,|\boldsymbol{x}-\boldsymbol{y}|)\leq \overline{B}_{n,s}e^{-\lambda|\boldsymbol{x}-\boldsymbol{y}|}.$$

These bounds would establish that the unified tempered fractional operator is asymptotically consistent with the tempered fractional Laplacian. Together with non-negativity of $F$ as a function of $|\boldsymbol{x}-\boldsymbol{y}|$, they would also support well-posedness of the corresponding exterior-value problem; the conjecture is supported by numerical evidence, while a full proof is left for future work.