MathNet 2023-part-04 — 191 problems (EN–VI) Question 101

For any positive integer $n$ let $n! = 1 \times 2 \times 3 \times \dots \times n$. Do there exist infinitely many triples $(p, q, r)$, of positive integers with $p \gt q \gt r \gt 1$ such that the product
$$p! \cdot q! \cdot r!$$
is a perfect square?