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

Let $A_{n,2}(x)$ be the rational polynomial part of the $n$th iterated integral in the $\ln(1+x^2)$ case, and let $\alpha_{n,2}$ be its denominator. Let $m\ge1$ be an integer. Denominator conjecture.

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

This is an arithmetic formulation of the predicted cancellations in the coefficients of $A_{n,2}(x)$, and the source gives no evidence that it has been proved or disproved.