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John - the disturbing thing about this is the range of responses you'll see to a question that in an "enterprise" grade product merits a DEFINITE answer. Even support's response is vague: "it doesn't really ask for your password so often."
What does "so often" even mean? That's like a car salesman saying the car may not start "every so often".
The reason for all this uncertainty is obviously that local connectivity with SPO is badly designed and does not work reliably. I and others have been saying that for 6 years and there are now 1000's of reports of issues caused by the flawed design which Msft continues to ignore. The most common official response is "try syncing" which of course comes with a whole other raft of problems and doesn't solve the issue at all.
Using Windows Explorer to open up SP locations is by far the best way to get users to adopt the product because of its familiarity. And the way to do it is to add network locations (right click "computer") for whatever sites/libraries you want to access. Generally once "trust" has been established, this will continue to work with no extra log in for a period -but no one seems to be able to define what that period is. You will eventually start seeing the "you're not authorized ...." or "invalid location ..." messages and will have to open up IE and click "open with explorer" again. Sometimes opening SP Designer can also trigger a reauthentication. If nothing works, there is not much you can do.
While I understand the need to harden authentication protocols for a product that could give someone access to a company's private content, you would think a co of Msft's size and expertise could do a better job of balancing that security need with usability. Right now, SPO sure doesn't seem like an "enterprise" grade product, does it?