  The article is concerned with the problem of the additivity of the tensor
rank. That is for two independent tensors we study when the rank of their
direct sum is equal to the sum of their individual ranks. The statement saying
that additivity always holds was previously known as Strassen's conjecture
(1969) until Shitov proposed counterexamples (2019). They are not explicit and
only known to exist asymptotically for very large tensor spaces. In this
article, we show that for some small three-way tensors the additivity holds.
For instance, we give a proof that another conjecture stated by Strassen (1969)
is true. It is the particular case of the general Strassen's additivity
conjecture where tensors are a pair of $2 \times 2$ matrix multiplication
tensors. In addition, we show that the Alexeev-Forbes-Tsimerman substitution
method preserves the structure of a direct sum of tensors.
