The octonion multiplication tensor's real subrank

Let $\mathbb O$ be the octonion algebra, viewed as an $8$-dimensional real algebra, and let $T_{\mathbb O}\in\mathbb R^8\otimes\mathbb R^8\otimes\mathbb R^8$ be its multiplication tensor. Octonion subrank conjecture. The real subrank of $T_{\mathbb O}$ equals $3$. The preceding discussion establishes only that this subrank is at most $4$, which is the generic subrank for real $8\times 8\times 8$ tensors; the stronger assertion that it equals $3$ is presented as a conjecture.