The generic tensor-rank conjecture for three-dimensional quaternionic factors

Let $T$ be a tensor in

$$\mathbb{H}^{2}\otimes_{\mathbb{H}}\mathbb{H}^{2}\otimes_{\mathbb{H}}\mathbb{H}^{2}.$$

The generic tensor-rank conjecture. The generic rank of $T$ is $3$. The preceding discussion suggests this quaternionic analogue of the complex generic-rank calculation, but does not establish it; the noncommutativity of the quaternions makes the corresponding geometric argument more difficult.