Conjectured asymptotics for characteristic-polynomial moments of the real Ginibre ensemble

Let $G_N$ be the $N\times N$ real Ginibre matrix, let $\gamma$ vary in a fixed compact subset of $(-\frac{1}{2},\infty)$, and let $x$ vary in a fixed compact subset of $(-1,1)$. The Barnes G-function is denoted by $G(z)$. Conjectured characteristic-polynomial moment asymptotics. The following asymptotic expansion holds uniformly for $\gamma$ and $x$ in these compact sets:

$$\mathbb{E}(|\det(G_N-x)|^{2\gamma}) \sim e^{-N\gamma(1-x^{2})}\left(\frac{N}{2}\right)^{\gamma^{2}-\frac{\gamma}{2}}\frac{(2\pi)^{\gamma}G(1/2)}{G(\gamma+1)G(\gamma+1/2)}, \quad N \to \infty.$$

This extends the preceding asymptotic formula at $x=0$ to characteristic polynomials evaluated at points in the bulk interval $(-1,1)$, with uniformity on compact parameter sets. The claim is motivated by the established result at $x=0$ and the corresponding integer-moment formula, but its general status is not resolved in the supplied text.