  Radio images of protoplanetary disks demonstrate that dust grains tend to
organize themselves into rings. These rings may be a consequence of dust
trapping within gas pressure maxima wherein the local high dust-to-gas ratio is
expected to trigger the formation of planetesimals and eventually planets. We
revisit the behavior of dust near gas pressure perturbations enforced by a
planet in two-dimensional, shearing box simulations. While dust grains collect
into generally long-lived rings, particles with small Stokes parameter $\tau_s
< 0.1$ tend to advect out of the ring within a few drift timescales. Scaled to
the properties of ALMA disks, we find that rings composed of larger particles
($\tau_s \geq 0.1$) can nucleate a dust clump massive enough to trigger pebble
accretion which proceeds to ingest the entire dust ring well within $\sim$1
Myr. To ensure the survival of the dust rings, we favor a non-planetary origin
and typical grain size $\tau_s \lesssim 0.05$--0.1. Planet-driven rings may
still be possible but if so we would expect the orbital distance of the dust
rings to be larger for older systems.
