Prove $\limsup_{n\to\infty}|\{(p,q)\in T\times T,p!q!=n\}|=6$

Question

Let $T$ be the set of nonnegative integers, I need to prove that $$\limsup_{n\to\infty}|\{(p,q)\in T\times T,p!q!=n\}|=6$$

It's really easy to show that

$$\limsup_{n\to\infty}|\{(p,q)\in T\times T,p!q!=n\}|\ge6$$ since for $n>2$ $$(n!)!=0!(n!)!=1!(n!)!=n!(n!-1)!=(n!-1)!n!=(n!)!1!=(n!)!0!$$

So I'm looking for a way to prove the apposite inequality.

Answer 1

This equality was proved by Daniel M. Kane in this paper.