Conjecture on the denominator ratios 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, let $\alpha_{n,2}$ be its denominator, and define

$$\beta_{n,2}=\frac{\alpha_{n,2}}{n\alpha_{n-1,2}}.$$

Here $p$ denotes a prime and $m,r\in\mathbb{N}$. The denominator-ratio conjecture. The sequence $\beta_{n,2}$ is given by

$$\beta_{n,2}=\begin{cases} p & \text{if $n=p^r$ for some \prime $p$ and $r\in\mathbb{N}$ and $n\ne2\cdot3^m+1$},\\ \frac13 & \text{if $n=2\cdot3^m$ for some $m\in\mathbb{N}$},\\ 3p & \text{if $n=2\cdot3^m+1$ and $n=p^r$ for some $m,r\in\mathbb{N}$},\\ 3 & \text{if $n=2\cdot3^m+1$ for some $m\in\mathbb{N}$ and $n\ne p^r$},\\ 1 & \text{otherwise.} \end{cases}$$

The claim describes the arithmetic cancellations in the denominators of the polynomial part. The source presents it as suggested by symbolic computations and gives no resolution evidence.