The universal-envelope conjecture for simple associative triple systems

Let $A_{p,q}$ ($p \le q$) be the simple associative triple system consisting of $(p+q) \times (p+q)$ block matrices of the form

$$\begin{bmatrix} 0 & p \times q \\ q \times p & 0 \end{bmatrix}.$$

Let $\omega$ be one of the following trilinear operations: Jordan ($q = 0, 1, \frac12$), anti-Jordan ($q = \infty, -1, 2$), the fourth family (all cases), cyclic commutator, weakly commutative, or weakly anticommutative. Universal-envelope conjecture. With finitely many exceptions, $U(A^\omega)$ is finite dimensional and semisimple and is isomorphic to

$$F \oplus M_{p+q}(F).$$

The conjecture arises from computations suggesting that the universal associative envelopes are generally finite dimensional and have only the trivial and natural irreducible representations. The finitely many exceptional cases and a general proof remain to be determined.