On the conjecture of the norm Schwarz inequality

For any positive invertible matrix $A$ and any normal matrix $B$ in $M_{n}({\Bbb C})$, we investigate whether the inequality $||A\sharp (B^{*}A^{-1}B)||\geq ||B||$ is true or not, where $\sharp$ denotes the geometric mean and $||\cdot||$ denotes the operator norm. We will solve this problem negatively. The related topics are also discussed.