Zero product of two bounded nonradial Toeplitz operators on the one-dimensional Fock space

Determine whether two nonzero bounded symbols on the one-dimensional Fock space F^2(C) can have a zero Toeplitz product T_fT_g=0.

Background

The paper constructs nonradial counterexamples in every dimension, but the one-dimensional construction yields a bounded second symbol only when the growth parameter satisfies c≥3/4; the first symbol is not established to be bounded in that construction. Consequently, the existence of two bounded nonzero symbols with zero Toeplitz product on F^2(C) is left unresolved.