$\mathbf{Question:}$
Let $u_{1},~u_{2} \in \mathbb{R}^{6}$ be two orthonormal vectors.
Let $\mathrm{C}_{1},~\mathrm{C}_{2},~\mathrm{C}_{3}$ be three real diagonal matrices with $\mathrm{C}_{1}^{2}+\mathrm{C}_{2}^{2}+\mathrm{C}_{3}^{2}=I_{6}$, where $I_{6}$ is a identity matrix. Then we can assume that \begin{align*} \mathrm{C}_{1}&=\mathrm{diag}(\cos a_{1},\cdots,\cos a_{6}),\\ \mathrm{C}_{2}&=\mathrm{diag}(\sin a_{1} \cos b_{1},\cdots,\sin a_{6} \cos b_{6}),\\ \mathrm{C}_{3}&=\mathrm{diag}(\sin a_{1} \sin b_{1},\cdots,\sin a_{6} \sin b_{6}). \end{align*}
Let $\mathrm{H}_{1}:=\mathrm{C}_{1} \left[ \begin{array}{cc} u_{1} & u_{2} \\ \end{array} \right]$, $\mathrm{H}_{2}:=\mathrm{C}_{2} \left[ \begin{array}{cc} u_{1} & u_{2} \\ \end{array} \right]$, $\mathrm{H}_{3}:=\mathrm{C}_{3} \left[ \begin{array}{cc} u_{1} & u_{2} \\ \end{array} \right]$. Then $\mathrm{H}_{1}^{T}\mathrm{H}_{1}+\mathrm{H}_{2}^{T}\mathrm{H}_{2}+\mathrm{H}_{3}^{T}\mathrm{H}_{3}=I_{2}.$
Let $\alpha_{1}\geq\beta_{1}\geq\gamma_{1}\geq\alpha_{2}\geq\beta_{2}\geq\gamma_{2}$ be the eigenvalues of $$D=\mathrm{H}_{1}\mathrm{H}^{T}_{1}+\mathrm{H}_{2}\mathrm{H}_{2}^{T}+\mathrm{H}_{3}\mathrm{H}_{3}^{T}.$$
$\mathbf{Prove}$ that the eigenvalues of $D$ satisfy the following inequalities: $$\alpha_{1}+\beta_{1}+\gamma_{2}\geq1,~\alpha_{1}+\beta_{2}+\gamma_{1}\geq1,~\alpha_{2}+\beta_{1}+\gamma_{1}\geq1.$$
$\mathbf{Idea:}$
Using Proposition 7 in the article ''Eigenvalues, invariant factors, highest weights, and Schubert calculus. Fulton, William. Bull. Amer. Math. Soc. (N.S.) 37(2000), no.3, 209–249.'', the necessary conditions about two conditions $\mathrm{H}_{1}^{T}\mathrm{H}_{1}+\mathrm{H}_{2}^{T}\mathrm{H}_{2}+\mathrm{H}_{3}^{T}\mathrm{H}_{3}=I_{2}$, $D=\mathrm{H}_{1}\mathrm{H}^{T}_{1}+\mathrm{H}_{2}\mathrm{H}_{2}^{T}+\mathrm{H}_{3}\mathrm{H}_{3}^{T}$ are obtained, but these conditions are not enough to prove the desired result.