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The article is concerned with the problem of the additivity of the tensorrank. That is for two independent tensors we study when the rank of theirdirect sum is equal to the sum of their individual ranks. The statement sayingthat additivity always holds was previously known as Strassen's conjecture(1969) until Shitov proposed counterexamples (2019). They are not explicit andonly known to exist asymptotically for very large tensor spaces. In thisarticle, 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 additivityconjecture where tensors are a pair of $2 \times 2$ matrix multiplicationtensors. In addition, we show that the Alexeev-Forbes-Tsimerman substitutionmethod preserves the structure of a direct sum of tensors.

Tensor rank of the direct sum of two copies of $2 \times 2$ matrix multiplication tensor is 14