  We propose a conjecture for exponential sums which generalizes both a
conjecture by Igusa and a local variant by Denef and Sperber, in particular, it
is without the homogeneity condition on the polynomial in the phase, and with
new predicted uniform behavior. The exponential sums have summation sets
consisting of integers modulo $p^m$ lying $p$-adically close to $y$, and the
proposed bounds are uniform in $p$, $y$, and $m$. We give evidence for the
conjecture, by showing uniform bounds in $p$, $y$, and in some values for $m$.
On the way, we prove new bounds for log-canonical thresholds which are closely
related to the bounds predicted by the conjecture.
