The characteristic-three classification conjecture for simple transposed Poisson algebras

Let $\mathbb{F}$ be an algebraically closed field of characteristic three. A simple finite-dimensional non-trivial transposed Poisson algebra over $\mathbb{F}$ has an associated Lie algebra, and let $\mathcal{W}(1;n)$ denote the Zassenhaus algebra; for $n\in\mathbb{N}$ and $q\in\mathcal{W}(1;n)$, let $\mathcal{W}_n(q)$ denote the corresponding transposed Poisson algebra.

Characteristic-three classification conjecture. Every simple finite-dimensional non-trivial transposed Poisson algebra over $\mathbb{F}$ is isomorphic to $\mathcal{W}_n(q)$ for some $n\in\mathbb{N}$ and $q\in\mathcal{W}(1;n)$. Equivalently, the associated Lie algebra of every such algebra should be a Zassenhaus algebra $\mathcal{W}(1;n)$.

The preceding classification applies in characteristic $p>3$; this conjecture asks whether the same description remains valid in characteristic three. Its resolution would complete the classification in the remaining prime characteristic considered here.