Argentium does tarnish, just more slowly. and if exposed to an
atmosphere with sulfur compounds, the process is sped up.
I invested in an ionic cleaner (Speed Bright) many years ago, and
have been pleased with its results. However, understand that as the
tarnish is removed, so are molecules from the surface. That means the
silver surface will eventually become dull, unless repolished.
May I suggest that you (also) consider a vibratory tumbler, such as
is used by reloaders to clean their brass. Just Google for "reloading
equipment" to find sources. The media is usually crushed walnut
shell or corn cob bits with some kind of charge. It is used dry. Run
the tumbler for an hour or so to clean off tarnish and polish the
metal. It's easy to check progress by pulling out a piece and
examining it. I have run pieces set with freshwater pearls and opals
with no damage to the stones - do check pieces with soft stones
frequently and pull them out at the earliest opportunity.
Judy in Kansas, where balmy temps were promised by the weather
forecaster. Certainly hope so!