Strassen's additivity conjecture for tensor rank

Let $\mathcal{S}^{\underline{n}}$ be the Segre variety for triple tensors, and let $\mathcal{S}^{(a_1,b_1,c_1)}$ and $\mathcal{S}^{(a_2,b_2,c_2)}$ be the Segre varieties associated with complementary subspaces in the three factors. For tensors $T_1\in \langle\mathcal{S}^{(a_1,b_1,c_1)}\rangle$ and $T_2\in \langle\mathcal{S}^{(a_2,b_2,c_2)}\rangle$, write $T_1\oplus T_2$ for their direct sum and $\operatorname{rank}$ for tensor rank. Strassen's conjecture. The rank is additive under direct sums:

$$\operatorname{rank}(T_1\oplus T_2)=\operatorname{rank}(T_1)+\operatorname{rank}(T_2).$$

This is Strassen's additivity problem for triple tensors. The source gives no resolution, so the conjecture remains open in the stated setting.