The dihedral and bicyclic Galois-group feasibility conjecture

Let $\mathcal{F}$ be the family of polynomial systems $F$ such that $Z(F)$ is finite and the Galois group of $F$ over $\mathbb{Q}$ is dihedral or bicyclic. Let $\operatorname{FEAS}_{\mathbb{C}}$ denote complex feasibility, namely the problem of deciding whether $Z(F)$ is nonempty. Dihedral and bicyclic feasibility conjecture. The restriction of $\operatorname{FEAS}_{\mathbb{C}}$ to $\mathcal{F}$ lies in $\mathbf{P}^{\mathbf{NP}^{\mathbf{NP}}}$ unconditionally. This proposes an unconditional upper bound for complex feasibility on systems whose zero set is finite and whose Galois group has the specified structure; the surrounding discussion presents it as an algorithmic analogue of the paper's results and as a conjecture based on an observation of Rachel Pries.