Two-dimensional diffusion orthogonal polynomials ordered by a weighted degree

We study the problem of describing the triples $(\Omega,g,\mu)$, $\mu=\rho\,dx$, where $g= (g^{ij}(x))$ is the (co)metric associated with a symmetric second-order differential operator $\mathbf{L}(f) = \frac{1}{\rho}\sum_{ij} \partial_i (g^{ij} \rho\,\partial_j f)$ defined on a domain $\Omega$ of $\mathbb{R}^d$ and such that there exists an orthonormal basis of $\mathcal{L}^2(\mu)$ consisting of polynomials which are eigenvectors of $\mathbf{L}$ and this basis is compatible with the filtration of the space of polynomials by some weighted degree.
In a joint paper of D. Bakry, M. Zani, and the author this problem was solved in dimension 2 for the usual degree. In the present paper we solve it still in dimension 2 but for a weighted degree with arbitrary positive weights.