Minimality conjecture for products of two independent non-centered normal random variables

Let $X$ and $Y$ be independent non-centered normal random variables, and let $W=XY$. A second-order Stein operator for $W$ is a differential operator acting on smooth functions with compact support, with polynomial coefficients, whose expectation under $W$ vanishes for every such test function. Minimality conjecture. There exists no second-order Stein operator for $W$ with polynomial coefficients acting on smooth functions with compact support. The paper presents this as an important open question concerning the minimal order of polynomial Stein operators; it also notes that the corresponding third-order operator is believed to be minimal, but that no proof is known.