Sporadic Wigner–Heisenberg algebra conjecture

Let $\hat{A}_1$ be the skew-hermitian Weyl algebra. For $n\geq 2$, let $\tilde{\mathcal{L}}_n$ denote the stated family of finite-dimensional solvable Lie algebras, and let $\mathfrak{r}(j_1,\ldots,j_n)$ denote the family indexed by integers $0\leq j_1<\cdots<j_n$. A finite-dimensional Lie subalgebra is non-nilpotent solvable if it is solvable but not nilpotent.

Sporadic Wigner–Heisenberg conjecture. The only finite-dimensional, non-nilpotent, and solvable Lie subalgebra of $\hat{A}_1$ that is not covered by the two families $\tilde{\mathcal{L}}_n$ or $\mathfrak{r}(j_1,\ldots,j_n)$ is the sporadic Wigner–Heisenberg algebra $\mathfrak{wh}_2$.

This conjecture asks whether the observed exception to the two known families is unique. The authors state that they have found no further examples in dimensions four or higher and leave the question for future work.