2- and 3-adic expansion conjecture for the gamma coefficients as alpha-polynomials

Let $r\in\mathbb{N}$ and set $a=\alpha^2-1$. Define $\gamma_{-\frac{1}{2}+r}^{\ast\ast}(a)$ by

$$\gamma_{-\frac{1}{2}+r}(a+1)=\frac{1}{(2r)!}\frac{a^r}{6^r}\gamma_{-\frac{1}{2}+r}^{\ast\ast}(a),\qquad \gamma_{-\frac{1}{2}+r}^{\ast\ast}\in\mathbb{Q}[a^{-1}].$$

2- and 3-adic expansion conjecture. The polynomials $\gamma_{-\frac{1}{2}+r}^{\ast\ast}(a)$ possess 2- and 3-adic expansions to all orders:

$$\gamma_{-\frac{1}{2}+r}^{\ast\ast}(a)=\sum_{0\leq j}\lambda_{2,j}(r,a)2^j,$$

where $\lambda_{2,j}(r,a)\in\{0,1\}$, and

$$\gamma_{-\frac{1}{2}+r}^{\ast\ast}(a)=\sum_{0\leq j}\lambda_{3,j}(r,a)3^j,$$

where $\lambda_{3,j}(r,a)\in\{0,1,2\}$. The coefficients depend only on the first $j$ digits of the corresponding $p$-adic expansion of $r$:

$$\lambda_{2,j}(r,a)=\lambda_{2,j}([r_0,r_1,\ldots,r_{j-1}],a),\qquad r=\sum_{0\leq i<j}r_i2^i\pmod {2^j},$$ $$\lambda_{3,j}(r,a)=\lambda_{3,j}([r_0,r_1,\ldots,r_{j-1}],a),\qquad r=\sum_{0\leq i<j}r_i3^i\pmod {3^j}.$$

Moreover, each $\lambda_{2,j}(r,a)$ has degree at most $j$ as a polynomial in $a^{-1}$.