Is it true that $\|A+PBP\|\le\|A+B\|$ for every projection $P$ and positive operators $A,B$? Let A and B be positive operators on and let P be a projection. Is the inequality
$$\|A+PBP\|\le\|A+B\|$$ true? Here $\|.\|$ stands for the operator norm.

Consider the case of $\mathbb C^2$ with the matrices
$$ A = \pmatrix{1 & 0\cr 0 & 6\cr},\ B = \pmatrix{6 & 0\cr 0 & 1\cr}, \ P = \pmatrix{1/2 & 1/2\cr 1/2 & 1/2\cr}$$
