Singly exponential generic decidability conjecture for Diophantine prefixes

A Diophantine prefix is generically decidable when an algorithm decides the corresponding positive-integer Diophantine sentences on the restricted collection of inputs specified by the source's genericity condition. Consider the prefixes $\exists\forall\exists$ and $\exists\exists\forall\exists$. Singly exponential generic-decidability conjecture. Both prefixes are generically decidable within singly exponential time. The source presents this as a further conjecture beyond its established generic decidability result; no resolution is given.