The dimension conjecture for simple null symplectic triple systems in characteristic 3

A symplectic triple system over a field of characteristic $3$ is called null when its alternating bilinear form is zero. Dimension conjecture. The dimension of any simple null symplectic triple system over a field of characteristic $3$ is two. The preceding discussion shows that every two-dimensional simple symplectic triple system in characteristic $3$ is null, so the conjecture would classify the dimensions of simple null systems in this characteristic.