Hypergeometric asymptotic expansion conjecture for $F(k,-\rho;k+n/2;z)$

Let $n$ be the dimension parameter, let $\rho=(n-1)/2$, and let $F(a,b;c;z)$ denote the hypergeometric function. For the function $F_k(z)=F(k,-\rho;k+n/2;z)$, with complex variable $k$ and fixed $z$, the asymptotic expansion conjecture.

$$F(k,-\rho;k+n/2;z)=(1-z)^{\rho}+\frac{(n/2)\rho}{k+n/2}\,z(1-z)^{\rho-1}+O(k^{-2}).$$

The expansion is intended as $|k|\to\infty$. The source notes that the assertion holds for even $n$ by a standard hypergeometric transformation, while for odd $n$ it is known only for $0\leqslant z<1/2$ and is otherwise unresolved there.