Non-existence of a short algorithm for multiplication of $3\times 3$ matrices whose group is $S_4\times S_3$, II

It is proved that there is no algorithm for multiplication of $3\times 3$ matrices of multiplicative length $\leqslant 23$ that is invariant under a certain group isomorphic to $S_4\times S_3$. The proof uses description of the orbits of this group on decomposable tensors in the tensor cube $(M_3(\mathbb{C}))^{\otimes 3}$ which was obtained earlier.