Explicit formula conjecture for the Poisson kernel $F_\gamma$

Let $\gamma$ be a non-negative integer, and let $F_\gamma$ be the Poisson kernel under consideration. For $1\leq\beta\leq\gamma+2$, let $f_\beta(x)$ be polynomials. Explicit $F_\gamma$ formula conjecture. The function $F_\gamma$ has the form

$$F_\gamma(z)=\sum_{\beta=1}^{\gamma+2}\frac{f_\beta(\lvert z\rvert^2)}{\lvert1-z\rvert^{2\beta}},\quad z\in\mathbb{D},$$

where

$$2f_\beta(x)=\sum_{k=0}^{\beta+\gamma+1-\max(2\beta-1,\gamma+2)}c_k\binom{\gamma+1-2k}{\beta-1-k}(1-x)^{\beta+\gamma+1-k}$$

and $c_0=1$, while

$$\sum_{k=0}^jc_k\binom{\gamma+1-2k}{j-k}=0$$

for $1\leq j\leq\lfloor(\gamma+1)/2\rfloor$. Equivalently, each $f_\beta(x)$ is a linear combination of powers $(1-x)^k$ with $\max(2\beta-1,\gamma+2)\leq k\leq\beta+\gamma+1$. The formula was verified computationally for $0\leq\gamma\leq80$, but the general assertion remains conjectural.