Highly unlikely. To emit 1 ton of CO2, you need to burn something like 111 gallons of gasoline. Average estimate of cost of externalities of CO2 emissions is around $50/tonne. That's less than 50 cents per gallon.
$50/ton is not a realistic carbon tax in 2022.
Yes, this is pretty high as far as tax is concerned, and unlikely to get introduced soon. However, it's still less than 50 cents a gallon. My point is that it's highly unlikely that "couple of dollars per gallon" is "a reasonable carbon tax", because carbon tax so high will be greatly exceeding the actual cost of CO2 emissions. "Couple of dollars per gallon" means at least $2/gallon, so $220/tonne CO2. That's enough to cover direct air capture of CO2, which is hardly the most cost-effective way of carbon capture.
I’ve seen claims that if you factor in negative externalities, the true price is at least double, and maybe nearer $15/gallon