Well, carbon electrodes don't dissolve fast enough in molten iron to keep them from being standard equipment for decades in EAFs, but you're right that that's not what they're using.

Scythe found this paper by Lan Yin, Antoine Allanore, and Sadoway from 02013 that says they're using 90% chromium, 10% iron, forming a refractory but conductive layer of eskolaite/alumina: https://pubmed.ncbi.nlm.nih.gov/23657254/ Boston Metal links this paper and calls out Allanore on their about page: https://www.bostonmetal.com/who-we-are/.

Sadoway is on Boston Metal's board, but Yin and Allanore evidently aren't involved.

Electric arc furnaces for steel don't typically immerse the electrodes in the molten metal (in the slag on top of the molten metal, yes.)

Oh, of course you're correct, thank you. How high is the solubility of graphite in molten iron?

Iron/carbon phase diagram: http://www.substech.com/dokuwiki/doku.php?id=iron-carbon_pha...

The eutectic of iron and carbon (mixture with the minimum melting point) is 4.3% carbon by mass.

There's a chapter from Richard Preston's "American Steel" (a 1992 book that I highly recommend about the Nucor minimill in Crawfordsville, Indiana) where they find at one point during the startup of the then-cutting-edge continuous casting line that the steel in arc furnace had too little carbon. They decided to solve that (expensively, for the purpose of the testing) by dipping the electrodes into the steel and letting their ends dissolve.