$\newcommand{\rank}{\mathop{\mathrm{rank}}}$Strassen conjectured for two tensors $T_{1}$ and $T_{2}$, $\rank(T_{1}\oplus T_{2})=\rank(T_{1})+\rank(T_{2})$. This is not generally true according to Yaroslav Shitov's paper. However since $\rank(T_{1}\oplus T_{2})\ge \max(\rank(T_{1}),\rank(T_{2}))$, we have $\rank(T_{1}\oplus T_{2})\ge\frac{1}{2}(\rank(T_{1})+\rank(T_{2}))$. My question is a weak form of this conjecture. I wonder if there is a constant $\epsilon>0$ such that for any $T_{1}$ and $T_{2}$, $\rank(T_{1}\oplus T_{2})\ge (\frac{1}{2}+\epsilon)(\rank(T_{1})+\rank(T_{2}))$.
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