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.
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.
However, the article suggests that actually the electrolyte might be neutral, containing both calcium and silicon oxides. That can't possibly be right if they're talking about only 1600° because larnite would precipitate out.
Electrodes would surely be carbon, just as they are in any old arc furnace. This does result in carbon dioxide emissions, as it does in the making of aluminum. Can they keep this down to an acceptable level? Is there an alternative electrode material, such as zirconia or carborundum? I'd like to know, but the article doesn't say.