Bertini's theorem for $F$-rationality is false

Let $k=\overline{{\mathbb{F}}_2}$. We construct a nine-dimensional $F$-rational affine variety $X$ admitting a locally closed embedding $X\hookrightarrow {\mathbb{P}}_k^{19}$ together with a dense open set $U\subseteq ({\mathbb{P}}^{19}{)}^{\vee }$ such that $X\cap H$ is not $F$-rational (or even $F$-injective) for all $H\in U$. We also construct a nine-dimensional projective $F$-rational variety $\mathfrak{X}$ and a closed embedding $\mathfrak{X}\hookrightarrow {\mathbb{P}}_k^{29}$ under which the same conclusion holds.