Fixed-point conjecture for the Berezin transform on Fock-type spaces

Let $f$ be a polynomial in $z$ and $\overline{z}$.

Fixed-point conjecture. If for any $m>0$ we have $B_mf=f$, then $f$ is harmonic.

The surrounding results establish this conclusion for polynomials of bounded degree outside discrete exceptional sets of parameters, and for all parameters in several special cases. It remains open whether the exceptional set in the general result can be non-empty.