Quark mass matrices inspired by a numerical relation

In 1981, Yoshio Koide noticed that the square root values of the charged lepton masses satisfy the relation $$Q=\frac{m_e+m_{\mu}+m_{\tau}}{(\sqrt{m_e}+\sqrt{m_{\mu}}+\sqrt{m_{\tau}})^2} \approx \frac{2}{3}$$ Inspired by this relation, we introduce tentative mass matrices, using numerical values, and find matrices that display an underlying democratic texture.