Finiteness conjecture for monic cubics with depth-one emergent reducibility

A monic cubic with depth-one emergent reducibility is a monic polynomial $f(x)\in\mathbb{Z}[x]$ of degree $3$ that is irreducible, while its first self-composition $f\circ f$ is reducible. Finiteness conjecture. There are only finitely many monic cubics in $\mathbb{Z}[x]$ with depth-one emergent reducibility. The known examples include several monic integral cubics with coefficients of absolute value less than $500$, while the conjecture asserts that this list is finite.