Converse direction of the Berger–Coburn endpoint characterization

Determine whether boundedness of the endpoint heat transform $g^{(1/4)}$ for an admissible symbol on the Fock space implies boundedness of the associated Toeplitz operator $T_g$.

Background

Berger and Coburn conjectured that, for an admissible symbol $g$, the Toeplitz operator $T_g$ is bounded if and only if its endpoint heat transform $g^{(1/4)}$ is bounded. The paper establishes that the forward implication fails by invoking Looi’s counterexample: a bounded Toeplitz operator can have an unbounded endpoint heat transform.

The unresolved issue is the converse implication: whether boundedness of $g^{(1/4)}$ is sufficient to ensure boundedness of $T_g$. The paper explicitly states that it does not address this direction, so the full endpoint characterization remains unresolved in that direction.