Let $k$ be a field. Let's say that $k$ has property $(∗)$ iff for a scheme $X$ and morphisms $f_1,f_2:X\rightarrow \mathrm{Spec}\,k$, geometric irreducibility of $f_2$ is implied by geometric irreducibility of $f_1$. Does there exist a non-pseudo-algebraically closed field with property $(∗)$?

any$X, f_1, f_2$, or whether there are two families $f_1, f_2$ depending on $X$ so that this is true, or something else? $\endgroup$ – user44191 Mar 13 at 10:12