Positive Semidefiniteness on off diagonal pertibation

Question

If $X$ is a positive semi-definite matrix and $Y$ is symmetric satisfying $X_{i,i}=Y_{i,i}$ and $ |Y_{i,j}| \leq |X_{i,j}| $ for all $i,j$ , is $Y$ necessarily positive semi-definite?

Answer 1

It is true for $n=2$ and false for $n>2$. Example:

$X=\begin{pmatrix}5554&5941&6859\\5941&12578&11157\\6859&11157&13914\end{pmatrix}$. To obtain $Y$, change $x_{3,1}$ and $x_{1,3}$ with their opposite.