Combining Igusa's conjectures on exponential sums and monodromy with semi-continuity of the minimal exponent.

We combine two of Igusa's conjectures with recent semi-continuity results by Musta\c{t}\u{a} and Popa to form a new, simple conjecture on bounds for exponential sums. These bounds have a formulation in terms of degrees and dimensions only. We provide evidence for our new question, partly by adapting already known results about Igusa's conjecture on exponential sums, but also some new evidence like for polynomials in 4 variables. We show that, in turn, these bounds imply consequences for Igusa's (strong) monodromy conjecture. The bounds lead to estimates for major arcs appearing in the circle method towards local-global principles; if minor arcs would follow this improvement (a huge challenge), then Birch's local-global principle for polynomials in many variables for given degree would generalize vastly.