Prime ideals and three-generated ideals with large regularity

Ananyan and Hochster proved the existence of a function $\Phi(m,d)$ such that any graded ideal $I$ generated by $m$ forms of degree at most $d$ in a standard graded polynomial ring satisfies $\mathrm{reg}(I) \le \Phi(m,d)$. Relatedly, Caviglia et. al. proved the existence of a function $\Psi(e)$ such that any nondegenerate prime ideal $P$ of degree $e$ in a standard graded polynomial ring over an algebraically closed field satisfies $\mathrm{reg}(P) \le \Psi(\mathrm{deg}(P))$. We provide a construction showing that both $\Phi(3,d)$ and $\Psi(e)$ must be at least doubly exponential in $d$ and $e$, respectively. Previously known lower bounds were merely super-polynomial in both cases.