An irregular smooth Fano fourfold in positive characteristic

We construct an explicit smooth projective Fano fourfold $X$ over an algebraically closed field $k$ of characteristic two with $H^1(X,{\mathcal{O}}_X)\simeq k$. Moreover, $X$ does not admit any free rational curve, and so it is not separably uniruled. The example has been constructed using ChatGPT.