Strassen's direct sum conjecture for tensor rank

Let $A_1,B_1,C_1,A_2,B_2,C_2$ be finite-dimensional vector spaces over a field $\mathbb{K}$, and let $T_1\in A_1\otimes B_1\otimes C_1$ and $T_2\in A_2\otimes B_2\otimes C_2$. Their direct sum is the tensor $T=T_1\oplus T_2$ in

$$(A_1\oplus A_2)\otimes(B_1\oplus B_2)\otimes(C_1\oplus C_2).$$

For a tensor $S$, write $\mathbf{R}(S)$ for its tensor rank, the minimum number of simple tensors whose linear combination equals $S$. Strassen's direct sum conjecture. The rank of the direct sum is the sum of the individual ranks:

$$\mathbf{R}(T)=\mathbf{R}(T_1)+\mathbf{R}(T_2).$$

The conjecture asserts rank additivity for independent tensors and, in particular, would identify the number of scalar multiplications needed for simultaneous matrix multiplication with the sum of the numbers needed for the two separate matrix multiplications. The source introduces this as a conjecture formulated by Volker Strassen in 1973; its resolution status is not established by the supplied text.