Hasler's ternary structural infinitude conjecture for binomial congruences

Let $n$ be a positive integer, and let $q$ and $p$ be primes with $p\neq q$. The binomial congruence is

$$\binom{qn}{n}\equiv q^n\pmod n.$$

Hasler's structural infinitude conjecture. The equation $\binom{3n}{n} \equiv 3^n \pmod{n}$ has infinitely many solutions of the form $n=3^k\cdot p$, where $k$ is a positive integer and $p$ is a prime number. More generally, the equation $\binom{qn}{n} \equiv q^n \pmod{n}$ has infinitely many solutions of the form $n=q^k\cdot p$, where $p\neq q$ are prime numbers and $k$ is a positive integer.

This conjecture proposes infinitely many prime-times-prime-power solutions, extending the observed ternary examples recorded in OEIS sequence A080469. The supplied text gives no resolution, so the conjecture remains open.