Positive integers: counterexample to W.M. Schmidt's conjecture

We show that there exist real numbers ${\alpha }_1,{\alpha }_2$ linearly independent over $\mathbb{Z}$ together with 1 such that for every non-zero integer vector $(m_1,m_2)$ with $m_1\ge 0$ and $m_2\ge 0$ one has $\mathrm{\mid }\mathrm{\mid }{m}_1{\alpha }_1+m_2{\alpha }_2\mathrm{\mid }\mathrm{\mid }\ge 2^{-300}(\max (m_1,m_2){)}^{-\sigma }$ with $\sigma ={1.94696}^{+}$.