Zero-free region conjecture for the hypergeometric function $F_k(z)$

Let $n$ be the dimension parameter, let $\rho=(n-1)/2$, and define

$$F_k(z)=F(k,-\rho;k+n/2;z),$$

where $F(a,b;c;z)$ is the hypergeometric function and $k$ is complex. The zero-free region conjecture. For every fixed relevant $z$, there exists $\varepsilon=\varepsilon(z,n)>0$ such that $F_k(z)$ has no zeros in

$$\{\operatorname{Re}(k)\geqslant -n/2-\varepsilon\}.$$

This conjecture is introduced to enable inversion, in the sense of Laplace transforms, of the function used in the paper. The supplied source gives no resolution or additional range in which it is known, so its status remains open.