  We combine two of Igusa's conjectures with recent semi-continuity results by
Musta\c{t}\u{a} and Popa to form a new, natural conjecture about bounds for
exponential sums. These bounds have a deceivingly simple and general
formulation in terms of degrees and dimensions only. We provide evidence
consisting partly of adaptations of already known results about Igusa's
conjecture on exponential sums, but also some new evidence like for all
polynomials in up to $4$ variables. We show that, in turn, these bounds imply
consequences for Igusa's (strong) monodromy conjecture. The bounds are related
to estimates for major arcs appearing in the circle method for local-global
principles.
