The Poisson-normalizer extension conjecture

Let ${\cal B}$ be a basic set with $\wp({\cal B})$ finite-dimensional. Let ${\cal N}(\wp({\cal B}))$ denote the Poisson normalizer of $\wp({\cal B})$.

Poisson-normalizer extension conjecture. Every integrable irreducible representation of $\wp({\cal B})$ can be extended to a quantization of

$$\big({\cal N}(\wp({\cal B})),{\cal B}\big).$$

Furthermore, no nontrivial quantization of this pair can be extended beyond ${\cal N}(\wp({\cal B}))$.

The conjecture is based on the cases of ${\bf R}^{2n}$ and $S^2$, where the relevant Poisson normalizers are identified with maximal consistently quantizable subalgebras. The general assertion is presented as a reasonable proposal and remains open in the source.