Berger–Coburn conjecture on the borderline heat transform criterion

Let $H^2(\mathbb C^n,d\mu)$ be the Bargmann–Fock space, let $g$ be measurable, and let $k_a$ denote the normalized reproducing kernel at $a\in\mathbb C^n$. For $t>0$, define the heat transform, whenever the integral is absolutely convergent, by

$$g^{(t)}(a)=(4\pi t)^{-n}\int_{\mathbb C^n}g(w)e^{-|w-a|^2/(4t)}\,dv(w).$$

Assume that $gk_a\in L^2(d\mu)$ for every $a\in\mathbb C^n$, and let $T_g$ be the associated Toeplitz operator. Berger–Coburn conjecture. $T_g$ extends to a bounded operator on $H^2(\mathbb C^n,d\mu)$ if and only if $g^{(1/4)}$ is bounded on $\mathbb C^n$.

The estimates of Berger and Coburn establish the corresponding norm inequalities for heat-transform times strictly above and strictly below $1/4$, leaving the endpoint $t=1/4$ as the unresolved case. The conjecture asks whether boundedness of this borderline transform is also sufficient for boundedness of the Toeplitz operator.