Uniqueness conjecture for faithful realizations of the Schrödinger algebra in the skew-hermitian Weyl algebra

Let $\mathcal{S}=\mathfrak{sl}(2,\mathbb{R})\ltimes\mathfrak{h}_1$ be the Schrödinger algebra, and let $\hat{A}_1$ be the skew-hermitian Weyl algebra with canonical commutation relation

$$[a,a^\dagger]=1.$$

Uniqueness conjecture. The Schrödinger algebra $\mathcal{S}$ admits a unique faithful realization as a subalgebra of $\hat{A}_1$, up to automorphism. This realization is the one explicitly constructed in Proposition~; no other subspace of $\hat{A}_1$ equipped with the canonical commutation relation is isomorphic to $\mathcal{S}$. The claim concerns the classification of finite-dimensional Lie subalgebras realized inside the Weyl algebra; the supplied text does not state whether this uniqueness assertion has been proved or remains open.