  In this article we conjecture a 4-dimensional characterization of tightness:
a contact structure is tight if and only if a slice-Bennequin inequality holds
for smoothly embedded surfaces in Yx[0,1]. An affirmative answer to our
conjecture would imply an analogue of the Milnor conjecture for torus knots: if
a fibered link L induces a tight contact structure on Y then its fiber surface
maximize Euler characteristic amongst all surfaces in Yx[0,1] with boundary L.
We provide evidence for both conjectures by proving them for contact structures
with non-vanishing Ozsv\'ath-Szab\'o contact invariant. We also show that any
subsurface of a page of an open book inducing a contact structure with
non-trivial invariant maximize "slice" Euler-characteristic for its boundary,
and conjecture that this holds more generally for open books inducing tight
contact structures.
