I would like to prove (or prove it is not true with a counter example) the following result:

Let $A$, $B$ be two squares matrices of size $n\times n$ with positive entries. If $A \leq B$, then $\sum_{i = 1}^n \sigma_i(A) \leq \sum_{i = 1}^n \sigma_i(B)$.

The sign $\leq$ between matrices is meant elementwise and the notation $\sigma_i(A)$ is used to denote the i-th singular value of $A$. The Perron-Frobenius theorem only provides information on the spectral radius, while the above statement involves all singular values.