Let $A$ and $B$ be a $n\times k_1$ and $n\times k_2$ matrices, and let $b$ be a vector in $\mathbb{R}^n$ (we may assume that $[1\ldots 1]^{\top}b=0$). We denote the projection matrix that sends a vector $v\in \mathbb{R}^n$ to $\operatorname{Col}^{\perp}(A)$ by $M_A$, i.e., $ M_A=I-A(A^{\top}A)^{-1}A^{\top}$ ($A$ has full column rank). Is there a vector $x\in\mathbb{R}^{k_2}$ with $\|x\|_2=1$ s.t. $$ \|B^{\top}M_A b\|_2\ge \|B^{\top} M_{[A\quad Bx]}b\|_2? $$ In addition, can we find $x$ with $\|x\|_2=1$ that minimizes the left-hand side?
The $M_{[A \quad v]}$ matrix is given by: $$ I-\left(AC^{-1}A^{\top}-\tilde{v}\tilde{v}^{\top}AC^{-1}A^{\top}-AC^{-1}A^{\top}\tilde{v}\tilde{v}^{\top}+\tilde{v}\tilde{v}^{\top}+\tilde{v}\tilde{v}^{\top}AC^{-1}A^{\top}\tilde{v}\tilde{v}^{\top}\right), $$ where $C=A^{\top}(I-\tilde{v}\tilde{v}^{\top})A$, and $\tilde{v}=v/\|v\|_2$.