  We prove an analogue for $p$-adic coefficients of the Deligne--Laumon theorem
on local acyclicity for curves. That is, for an overconvergent $F$-isocrystal
$E$ on a relative curve $f:U\rightarrow S$ admitting a good compactification,
we show that the cohomology sheaves of $\mathbf{R}f_!E$ are overconvergent
isocrystals if and only if $E$ has constant Swan conductor at infinity.
