Let $A$ and $B$ be two $n\times n$ hermitian matrices. Does $U^{*}AU+B \prec_{w} A+B$ for any unitary matrix $U$? Here the notation $``\prec_{w}"$ stands for the weak majorization, that is, $x\prec_{w} y$ if and only if $\sum\limits_{j=1}^{k}\lambda_{j}^{\downarrow}(x)\leq \sum\limits_{j=1}^{k}\lambda_{j}^{\downarrow}(y)$, for each $1\leq k\leq n$, and $\{\lambda_{j}^{\downarrow}(x)\}_{j=1}^{n}$ and $\{\lambda_{j}^{\downarrow}(y)\}_{j=1}^{n}$ are eigenvalues of $x$ and $y$ ordering in the non-increasing order respectively.