Is there a degree-5 polynomial $ p\left(t\right) = a_0 + a_1 \cdot t + a_2 \cdot t^2 + a_3 \cdot t^3 + a_4 \cdot t^4 + a_5 \cdot t^5 $ with integer coefficients $ a_j $, such that $ p\left(t\right) $ is irreducible over rationals, but the second iterate $ p\left(p\left(t\right)\right) $ is not irreducible over rationals?

This is a special case of an earlier question I asked on Math.SE. According to an answer there, such polynomials are known as polynomials with newly reducible second iterate. The same answer by Sil gives this example of a degree-5 polynomial with rational coefficients: $$ \begin{align} p\left(t\right) &= t^5 + 10 t^4 + 40 t^3 + 80 t^2 + \frac{1232}{15} t + \frac{15904}{465} \\ p\left(p\left(t\right)\right) &= \left( t^5 + 10 t^4 + 40 t^3 + 80 t^2 + \frac{1202}{15} t + \frac{14974}{465} \right) \cdot Q\left(t\right) \end{align} $$ where $ Q\left(t\right) $ is shorthand for a degree-20 factor.

I searched for an example with integer coefficients using a script, and found no examples where $ \sum_j {a_j^2} < 100 $.