When is it true that $(y_1-x_1)^p + (y_2 -x_2)^p \leq (y_2 - x_1)^p + (y_1 - x_2)^p$?

Question

Consider four integers $x_1, x_2, y_1, y_2$ with $x_2 > x_1$ and $y_2 > y_1$. Is it true that the only integer values of $p$ so that:

$$(y_1-x_1)^p + (y_2 -x_2)^p \leq (y_2 - x_1)^p + (y_1 - x_2)^p$$

are $p=0, 1$ and all even $p \geq 2$?