Proving a matrix inequality involving a extended matrix and the pseudoinverse under the background of signal processing

Question

Consider two positive definite Hermitian matrices $\pmb W_1$ and $\pmb W_2$ such that $\pmb W_1 \succ \pmb W_2$, which means $\pmb W_1-\pmb W_2$ is positive definite. There exist two full cloumn rank matrices $\pmb A_1$ and $\pmb A_2$ with more rows than columns. The number of rows in $\pmb A_1$ and $\pmb A_2$ are the same, while no zero rows are included. Moreover, assume $\pmb A_2=[\pmb A_1,\pmb a]$, where $\pmb a$ is a non-zero vector. Then let $\pmb B_1=\pmb W_1\pmb A_1$ and $\pmb B_2=\pmb W_2\pmb A_2$. I am seeking the proof of the following inequality $$\pmb W_1^H(\pmb I-\pmb B_1\pmb B_1^+)\pmb W_1 \succ \pmb W_2^H(\pmb I-\pmb B_2\pmb B_2^+)\pmb W_2.$$ Here $\pmb B_1^+$ refers to $\pmb B_1^+=(\pmb B_1^H\pmb B_1)^{-1}\pmb B_1^H$, and $\pmb I$ is the identy matrix. Note that, in the context of signal processing, $\pmb A_1$ and $\pmb A_2$ are the so-called steering matrices, whose elements are all complex exponential. $\pmb W_1$ and $\pmb W_2$ can be expressed by $$\pmb W_1=(\pmb R_1^T \otimes \pmb R_1)^{-1/2},\\\pmb W_2=(\pmb R_2^T \otimes \pmb R_2)^{-1/2},$$ where $\otimes$ denotes the Kronecker product. $\pmb R_1$ and $\pmb R_1$ stand for the signal co-variance matrices, and $\pmb R_2 \succ \pmb R_1$.