Conjecture on the auxiliary denominators for the iterated integral of $\ln(1+x^2)$

Let $G_n(x)$ be the auxiliary polynomial introduced by expanding the expression for $A_{n,2}(x)$, and let $\gamma_n$ be its denominator. Auxiliary-denominator conjecture. For $n\ge2$,

$$\gamma_n=\begin{cases} \dfrac{\operatorname*{\lcm}(1,2,\dots,n)}{6} & \text{if $n=2\cdot3^m$ for some $m\ge1$},\\ \dfrac{\operatorname*{\lcm}(1,2,\dots,n)}{2} & \text{otherwise.} \end{cases}$$

By the preceding corollary, this is equivalent to the conjectured formula for $\alpha_{n,2}$, so it is a restatement rather than a separate mathematical claim.