Algebraic Stein operator characterization for Gaussian polynomials

Let $X\sim N(0,1)$), and let $Y=h(X)$, where $h\in\mathbb{K}[X]$ is a polynomial with $\deg(h)\geq 3$ and ${\rm I\kern-0.16em E}[h(X)]=0$. For $p_0\in\mathbb{K}_0[Y]$, let $\Sigma=(\mathbb{K}[X],\mathbb{K}[Y],\Gamma_Y)$ be the associated linear machine, and let $\mathcal{N}(\Sigma,T)$ denote the corresponding space. The operator composition is written $\Gamma_Y^T$. Algebraic Stein operator characterization. For every $p_0\in\mathbb{K}_0[Y]$, the following equivalent statements hold: (a) there exists a natural number $T\geq2$ such that $Y$ admits an algebraic polynomial Stein operator $\mathcal{S}$ of order $T$ with zero-order polynomial coefficient $p_0$; (b) there exists a natural number $T\geq2$ such that $p_0\in\mathcal{N}(\Sigma,T)$; and (c) there exists a natural number $T\geq2$ such that

$$\Gamma_Y^T(p_0)\in\sum_{t=1}^{T-1}\Gamma_Y^t(\mathbb{K}[Y]).$$

This conjecture characterizes the existence of algebraic polynomial Stein operators for polynomial transformations of a standard Gaussian random variable in terms of the associated linear machine and iterated operators. The degree-$1$ and degree-$2$ cases are referred to earlier results, while the stated characterization for degree at least $3$ is presented as an open problem.