The quasi-irreducibility criterion for holomorphic eta quotients

Let $f$ be a holomorphic eta quotient of level $N$. It is quasi-irreducible if it is not factorizable on $\Gamma_0(N)$, whereas it is irreducible if its only factors are $1$ and $f$. Quasi-irreducibility conjecture. Every quasi-irreducible holomorphic eta quotient is irreducible. This is the contrapositive of the reducibility conjecture and would provide a direct algorithmic test for irreducibility at the given level; the source gives no resolution.