I think that particular claim is probably true, or at least close.

Most claims like this I've seen are based on the Edict of Diocletian, a series of maximum price controls passed by the Roman emperor in response to persistent inflation despite monetary reform. These are dictated prices, not market prices. Nonetheless they do tend to roughly mirror current prices. I would expect the metal price of agricultural goods and basic labor in the late pre-industrial age to closely mirror Roman prices.

I don't believe in a general principle that metal purchasing power remains flat. I think with technological progress, capital accumulation & expanded division of labor real costs should fall. Looking at USA through the 1800s for example I believe we see mild price deflation over the century.

In contrast, capital atrophy can raise prices, especially making worse an already-inflationary regime.

We also know of certain periods where precious metal supply suddenly spiked, producing temporary price inflation & decrease of real purchasing power. Roman emperor Trajan acquired silver mines in Spain producing temporary price inflation in silver, and the California gold rush produced temporary price inflation in gold.

However, generally yes, precious metals maintain purchasing power well over long periods. The rate of dilution (from mining) is usually quite modest & roughly mirrors population growth. There are also multiple natural balancing effects. If the metal has high purchasing power, mines can more readily afford mine employees & equipment, producing more metal. Conversely mine activity & metal supply stagnates when the purchasing power of the metal is low. Another balancing effect comes from the fact that precious metal mining can't really be much more efficient than other forms of mining (or even other industrial activity in general). So, the availability of precious metal tends to closely parallel the availability of other commodities, producing stable relative prices.