Is it true that $\min\left(\lambda_{\min}(M_{12}),\lambda_{\min}(M_{13}),\lambda_{\min}(M_{23})\right) \le \frac{2}{5}$ where $M = \mathbf{x}_{1}\mathbf{x}_{1}^{\top} + \mathbf{x}_{2}\mathbf{x}_{2}^{\top},$ for all $\mathbf{x}_{1}, \mathbf{x}_{2} \in \mathbb{R}^{3}$ with $\|\mathbf{x}_{1}\| = \|\mathbf{x}_{2}\| = 1$?
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