That is, is it true that the bound $$\phi(mn)\leq m\phi(n)$$ holds for all pairs of natural numbers $m$ and $n$?

It is true on average, in the sense that $$\sum_{mn=k}m\phi(mn)\leq\sum_{mn=k}mn\phi(n),$$ and it holds for every pair I have computed. If it does not hold, what is the smallest counter example?