Sufficiency of the necessary condition for nice integers

Prove that if the smallest prime divisor p of a positive integer n is such that n/p has fewer than p distinct prime divisors, then the divisors of n greater than 1 are the moduli of a good congruence set; equivalently, establish that the condition in Lemma 1 is sufficient for n to be nice.

Background

The paper defines an integer n to be nice if there are residue classes with moduli given by all divisors d>1 of n such that any two overlapping congruences have coprime moduli. Lemma 1 proves a necessary condition: if p is the smallest prime divisor of n, then n/p has fewer than p distinct prime divisors. The paper verifies niceness for some families, including prime powers and certain products of a prime with a prime power, but does not establish the converse in general. The conjecture asks whether the necessary condition completely characterizes nice integers.