Fischer operator bijectivity conjecture for polynomials

Let $P$ be a polynomial on $\mathbb{C}^{d}$, let $E(\mathbb{C}^{d})$ denote the space of entire functions, and define the Fischer operator

$$F_P:E(\mathbb{C}^{d})\to E(\mathbb{C}^{d}),\qquad F_P(q)=P^{\ast}(D)(Pq).$$

Fischer operator bijectivity conjecture. The operator $F_P$ is a bijection. The source states that this conjecture remains open in general, although it records several special cases and partial results.