An exercise in number theory

Here is a cute little problem which I can’t solve. I thought I needed the answer but it turned out that I didn’t, so as far as I know there is no application.

Let n = p₁^a₁…p_s^a_s, where the p_i are distinct primes and a_i positive integers. The general question asks whether φ(n) is always at least as big as the multinomial coefficient (a₁+…+a_s)!/(a₁)!…(a_s). This is true if s = 1 or 2, false if s = 4 (or larger) – the smallest example I know is 2⁷.3⁵.5³.7² = 190512000.

Problem: Is it true for s = 3? I am inclined to guess that it is, but it is quite delicate and I didn’t find a proof. (I have stopped looking now I don’t actually need this any more.)