Berger–Coburn conjecture on bounded Toeplitz operators on Fock spaces

Let $t>0$ and let $f$ be a symbol for which the Toeplitz operator $T_f^t$ and its Berezin transform $tilde{f}^{(t/2)}$ are defined on the $t$-Fock space. Berger–Coburn conjecture. The operator $T_f^t$ is bounded if and only if

$$\widetilde{f}^{(\frac{t}{2})}$$

is bounded. This conjecture was made by C. Berger and L. Coburn in the case $t=\frac{1}{2}$; the general boundedness characterization remains open in the source.