The OP already gave the series expansion for $\arctan \left( \frac{2x \sin\theta}{1-x^{2}} \right) $ in here. Using that expansion, we have
\begin{align*}
&\int_{0}^{\frac{\pi}{2}} \arctan \left( \frac{2x \sin\theta}{1-x^{2}} \right) \arctan \left( \frac{2y \sin\theta}{1-y^{2}} \right) \arctan \left( \frac{2z \sin\theta}{1-z^{2}} \right) \arctan \left( \frac{2w \sin\theta}{1-w^{2}} \right) \, d\theta \\
&\quad = 16 \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \sum_{p=0}^{\infty} \sum_{q=0}^{\infty} \frac{x^{2m+1}y^{2n+1}z^{2p+1}w^{2q+1}}{(2m+1)(2n+1)(2p+1)(2q+1)} \times \\
& \qquad \times \int_{0}^{\frac{\pi}{2}} \sin(2m+1)\theta \sin(2n+1)\theta \sin(2p+1)\theta \sin(2q+1)\theta \, d\theta \quad \quad \rm{\bf (1)}
\end{align*}
Now we use the following somewhat lengthy trigonometric equality for general $x,y,z,w$ (which can be shown by Euler formulae, or more readily, by combining triple and double product formulae as listed in Wikipedia here)
\begin{align*}
& \quad 8 \sin(w) \sin(x) \sin(y) \sin(z) = \\
& - \cos(w+x+y-z) - \cos(w+y+z-x) - \cos(w+z+x-y) + \cos(w+x+y+z)+\\
&\cos(-w+x+y-z) + \cos(-w+y+z-x) + \cos(-w+z+x-y) - \cos(-w+x+y+z)
\end{align*}
Integrals arise, where 4, 3, or 2 arguments have a positive sign. Let's look at these type of integrals. In the following, $\delta_n = 1$ for $n=0$, $\delta_n =0$ else.
a) 4 positive signs:
$$
\int_{0}^{\frac{\pi}{2}} \cos(2m+2n+2p+2q+4)\theta \, d\theta = \frac{\pi}{2} \delta_{m+n+p+q +2}
$$
Since $m,n,p,q \geq 0$, this term will never be met in the sum.
b) 3 positive signs:
$$
\int_{0}^{\frac{\pi}{2}} \cos(2m+2n+2p-2q +2)\theta \, d\theta = \frac{\pi}{2} \delta_{m+n+p-q +1}
$$
c) 2 positive signs:
$$
\int_{0}^{\frac{\pi}{2}} \cos(2m+2n-2p-2q)\theta \, d\theta = \frac{\pi}{2} \delta_{m+n-p-q}
$$
So the full integral (1) becomes
\begin{align*}
& \frac{\pi}{16}\Big[ \delta_{m+n+p+q +2} - \delta_{-m+n+p+q +1}- \delta_{m-n+p+q +1}- \delta_{m+n-p+q +1}- \delta_{m+n+p-q +1} +\\
& \delta_{m+n-p-q} +\delta_{-m+n-p+q} +\delta_{m-n-p+q} \Big]
\end{align*}
So the conjecture is proved if we establish
\begin{align*}
& \frac12 (-1)^{m+n+p+q} d(m,n,p,q) = \quad \quad \rm{\bf (2)}\\
& \delta_{m+n+p+q +2} - \delta_{-m+n+p+q +1}- \delta_{m-n+p+q +1}- \delta_{m+n-p+q +1}- \delta_{m+n+p-q +1} +\\
& \delta_{m+n-p-q} +\delta_{-m+n-p+q} +\delta_{m-n-p+q}
\end{align*}
Since $d(m,n,p,q)$ denotes the number of choices of signatures so that
$$
\pm(2m+1) \pm(2n+1) \pm (2p+1) \pm(2q+1) = 0
$$
we can convince ourselves of the various cases. This is best seen with the case description given by the OP. (Note my comment above, there appears to be a typo in the one but last case.) Here is a list of terms (with corresponding signs) which apply. Note that we need half as many terms as given by the OP, due to the factor $\frac12$ in (2):
$$
\begin{cases}
\delta_{m+n-p-q} +\delta_{-m+n-p+q} +\delta_{m-n-p+q} & \text{if } m = n = p = q, \\
\delta_{-m+n-p+q} +\delta_{m-n-p+q} & \text{else if } m = n < p = q, \\
\delta_{m+n-p-q} & \text{else if } m + n = p + q, \\
- \delta_{m+n+p-q +1} & \text{else if } m + n + p = q-1, \\
0 & \text{else}.
\end{cases}
$$
This holds as well for all permutations of the indices. Done. $\quad \quad \Box$
