Zero-product problem for pluriharmonic symbols of exponential type

Determine whether pluriharmonic symbols f=f_1+\overline{f_2} and g=g_1+\overline{g_2} on F^2(C^n), whose entire parts f_1,f_2,g_1,g_2 have exponential growth, must satisfy f=0 or g=0 whenever T_fT_g=0 on F^2(C^n).

Background

The paper proposes a restricted class in which the general counterexamples may not apply: pluriharmonic symbols whose entire components have exponential type. In one complex dimension, a prior result establishes the implication when both ordered products vanish, T_fT_g=T_gT_f=0. The unresolved question asks whether the stronger two-sided hypothesis can be reduced to the single condition T_fT_g=0 and whether the resulting implication holds in every complex dimension.