Vibratory tumbling of sterling chains

I have a lot of tarnished sterling chains. I’m wondering is there is
a way to tumble them with good results. I’ve used things like Tarn-X
on them, but to me that just makes them dull. This is just tarnish,
no body oils, as they have never been worn. I am having good luck
removing tarnish on pendants with dry corn cob with cerium oxide.
But comparatively speaking, most of the chains are much more
tarnished than the pendants.

(FWIW : I do have an ionic cleaning machine: I have given up using
it, as it seems quite ineffective. It also seems to leave pieces
dull, and still needing a final polish (mine does not have
agitation: they are not made with agitation anymore). Whereas the
vibratory tumbler is doing a bang-up job, and can do lots of pieces
in one load. )

Most the chains are box or snake. Obviously the tumbler can’t get
into the inner link surfaces. Has anyone tried Tarn-X to get that
tarnish off, then rinse, dry (completely), then tumble?

I also have Rio Red Buff and Green Buff. I have not tried either of
them yet.

Thanks!