Would be very interesting to see what happens with the K24. Mine also burns about a quart every 1k or so, and has like clockwork since I bought it in 2016. Granted, it has a lot more miles than the one you're testing, but it also hasn't gotten any worse. I've tried many different brands, and thicker grades to no avail.
Here's what kind of worked on our K24. Now this is the typical "throw crap at a wall and see what sticks." I am fully owning the fact this means nothing, and it could just be luck at play, or nothing, one thing, or the synergy of things.
Tried and not too helpful:
5W-30 oils
Any kind of high mileage oil
Tried and moderately helpful:
1. Ran a full bottle of Berryman B60 through it after a fresh oil change with Kirkland 0W-20. Ran it 3,000 miles. Changed oil and filter. Slight reduction in oil consumption.
2. Cleaned out PCV. It was working, but cleaned it with some gun carbon cleaner.
3. Did a change of Valvoline R&P with fresh filter after number 1. Used 5W-30 since it's hot, and we were doing a lot of family trips on the highway at high speeds with tons of crap in the car so this pathetic engine under high load. So far 2000 miles on that. The engine oil always smells of fuel, more so than my other Hondas, so I just don't have confidence it's not diluting the oil with unburned fuel hence the 5w-30.
Over 6 months, the combination of 1+3 did seem to help mitigate oil consumption from the 1 liter per 1k miles down to about 500ml in 1500 miles. I don't expect it will ever be "fixed." It's high mileage. The engine seems to be mostly content with the 30W oils.
Fuel economy over this period went from 19mpg (below EPA rating) to 22mpg (still slightly below) with identical driver and driving.
I don't really see the cost benefit to doing a lot of oil testing and diagnosis on this since I don't want to pour a bunch of money into an old car approaching 170,000 miles. I'll try the mechanic in a bottle if the price is right and the risk is low. But I'm at the point where I don't see much else to throw at this. Might as well be a sucker for marketing and do 20,000 on the R&P doing 5k changes to see what happens.