It varies on my server. Yew almost always has more supply than demand and I've seen prices drop as low as 4.5k before. Walnut goes through phases of too many and none up. It's rare that there's none up however, but it depends how many botanists are logging that week. Now imagine all the walnut logs ever to exist and add 10% of that figure on, that's how many logs would exist in the wards right now. Demand won't eat them up because that would just reduce demand from new logs coming about. Like I say it's rare that there's 0 up and eventually (usually within a few days) supply is going to catch up to the current demand. I've had stacks go unsold for over a week sometimes.
All other logs except mahogany suffer from massive over supply issues because they aren't skilled up on. No one needs 500 pine or chestnut logs. Why mahogany doesn't suffer from this is because the cap for chopping them is 1 per chop and it's a hard log to chop. This is exactly why I believe everything should be capped at 1. Those other logs would have value. But sure, yew and walnut may suffer supply issues then. The solution is more walnut and yew trees in grade 5. Or with a cap of 1 you could perhaps use your idea and make walnut easier to gather with the right sweetspot.