The generalized native-space inclusion conjecture

Let $\mathbf{P}$ and $G$ satisfy the conditions of the theorem identifying the native space with $\mathrm{H}_{\mathbf{P}}(\mathbb{R}^d)$. Let $\mathscr{P}$ be a finite-dimensional subspace of the null space of $\mathrm{H}_{\mathbf{P}}(\mathbb{R}^d)$ such that

$$\pi_{m-1}(\mathbb{R}^d)\subseteq\mathscr{P}.$$

Generalized native-space inclusion conjecture. The native space $\mathcal{N}_G^{\mathscr{P}}(\mathbb{R}^2)$ associated with $G$ and $\mathscr{P}$ is a subspace of $\mathrm{H}_{\mathbf{P}}(\mathbb{R}^d)$.

This is proposed as an extension of the paper's results on generalized Sobolev native spaces, motivated by the modified thin plate spline example. The statement concerns the relationship between a native space defined using a larger finite-dimensional null-space subspace and the generalized Sobolev space; no resolution is given in the supplied text.