Finitary simplicity of the limiting dynamical symmetry Lie algebra

Let $\Phi[\sigma;1;f]$ be the limiting object considered in the paper, and let $(\mathfrak l_{k_h})_h$ be the system of dynamical symmetry Lie algebras associated with the embeddings $\mathfrak l_{k_h}\hookrightarrow\mathfrak l_{k_{h+1}}$. Finitary simplicity conjecture. The Lie-algebra direct limit of this system is a finitary simple Lie algebra. In the notation used in the source, the subsequent classification identifies the direct limit with $\mathfrak{sl}_\infty(\mathbb C)\cong\mathfrak{fsl}(\mathscr H_\sigma)$, where $\mathscr H_\sigma=(L^2_\sigma)_{\mathbb C}$, and its norm closure with the trace-zero compact operators $\mathscr K_0(\mathscr H_\sigma)$. This is motivated by the classification of finitary simple Lie algebras and is used to describe the limiting dynamical symmetry algebra; the supplied text does not state whether the conjecture has been resolved.