Do we know this yet? It seems awfully unlikely that we couldn't have a pretty solid answer to this question by now, given how many Omicron cases there have been by now.

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I think ZVI is uncharacteristically too conservative here. The data from South Africa is becoming overwhelming. They are on the downside of the peak and simply haven't seen any substantial increase in deaths. I think we are in the 98% range for overall Omicron cases being much less lethal (including vaccine breakthrough) and probably in the 90% range for it being less lethal only considering unvaccinated cases.

My best guess is the death toll in US for this wave is 1/4 of Delta wave despite far more cases.

Zvi's Omicron summary is probably your best source of information:

Omicron probably milder than Delta (~50%) so baseline IFR likely ~0.3% unless hospitals overload, lower for vaccinated or reinfected.