Galois groups of all Catalan truncations

Determine whether the Galois group over \(\mathbb{Q}\) of every truncation \(P_n(x)=\sum_{k=0}^{n} C_kx^k\) of the Catalan generating function is isomorphic to the symmetric group \(S_n\) for every positive integer \(n\).

Background

For the Catalan truncations $P_n(x)=\sum_{k=0}^{n}C_kx^k$, the paper proves irreducibility over $\mathbb{Q}$ for every positive integer $n$. Computational work had previously verified that the Galois group is $S_n$ for $1\leq n\leq 2000$. The stronger assertion that this holds for every positive integer $n$ is attributed to a conjecture of Zhi-Wei Sun and is explicitly left unaddressed by the paper.