Classification of two-dimensional Leibniz triple systems

Let $F$ be an algebraically closed field of characteristic $0$, and let $T$ be a $2$-dimensional Leibniz triple system over $F$ with product $\langle-,-,-\rangle$. The systems in question are the five systems listed as systems 1–5, with the fifth depending on a parameter $\zeta\in F$.

Classification conjecture. Every $2$-dimensional Leibniz triple system over $F$ is isomorphic to one of the systems 1–5.

The statement proposes a complete classification of the two-dimensional Leibniz triple systems over an algebraically closed field of characteristic $0$. The supplied text presents the five systems as solutions, but gives no evidence of a proof or resolution of the asserted completeness.