The automorphism-group conjecture for algorithms computing matrix multiplication

Let $\mathcal{A}$ be an algorithm of length $r$ computing the matrix multiplication tensor $\langle m,n,p\rangle$. Its automorphism group is denoted by $Aut(\mathcal{A})$. Let $\mathcal{P}_m$, $\mathcal{P}_n$, and $\mathcal{P}_p$ be the corresponding groups of generalized permutation matrices, and let $Q(m,n,p)$ be the group appearing in the natural semidirect-product action on algorithms. Automorphism-group conjecture. Suppose that $\mathcal{A}$ is an algorithm of any length $r$ computing $\langle m,n,p\rangle$. Then

$$Aut(\mathcal{A})\leq(\mathcal{P}_m\times\mathcal{P}_n\times\mathcal{P}_p)\rtimes Q(m,n,p).$$

The conjecture asserts that the $D$-property and weak $D$-property used in the preceding partial results are unnecessary. It is intended to apply to algorithms of arbitrary length, whether $r$ is less than, equal to, or greater than $mnp$; the supplied source does not state a resolution.