The zero-product conjecture for Toeplitz operators on Fock-Sobolev spaces

Let $m\neq 0$, let $f,g,h,k\in\mathcal{A}_1$, and define

$$u=f+\overline{k},\qquad v=h+\overline{g}.$$

Here $T_u$ and $T_v$ are the Toeplitz operators with symbols $u$ and $v$ on the Fock-Sobolev space $F^{2,m}$. Zero-product conjecture. If

$$T_uT_v=0$$

on $F^{2,m}$, then either $u=0$ or $v=0$.

The conjecture relaxes the previously considered hypothesis requiring both products to vanish. Its validity is not established in the supplied text.