If $n = 18k+5$ is composite, there are at least 9 divisors of $\phi(n)$ which do not divide $n-1$.

Question

If $n$ is a composite of the form $18k+5$, there at least 9 divisors of $\phi(n)$ which do not divide $n-1$. Is this true in general or if not, what is the smallest counter example?

Update: No counter example for $n \le 1 \times 10^7$.

Related question.

Note: Posted in MO since it did not get an answer in MSE