Obviously not.If it's compatible with 5e, then nobody will buy it - they'll just use the 5e books they already own. That means that Pathfinder as a game is done.
1. Making a 5e-based PF system require the 5e-based PF core book would be absolutely trivial, if that's what Paizo wants. I'm amazed I even have to say this (but in case I did: new sub-classes or classes, new spells, bit of new equipment, few new feats, i.e., do your job, and you have a book that subsequent products can reference, thus requiring players to buy the core book).
2. A 5e PF will have to have plenty of dissimilarities, just to fill out a PF-sized core rulebook. WotC trimmed quite a bit out of the PHB for the SRD, so they'll have to be different from official 5e, by definition.
3. We're talking about one book out of a whole line anyway. Meanwhile, Paizo could start selling 5e-based (insert Paizo marketing euphemism for PHB2, MM2, MM3, splatbook galore, etc., here.) to WotC's 5e customers, as well as their established Pathfinder customer base.
4. If Paizo wants to compete with the PHB/DMG specifically, they can do so. If they want to piggy-back onto 5e's core books, they can do that, too. The choice is theirs. If they want to compete, they can release in a different format (all core in one book, like now), with different rules (to replace all the stuff WotC held back from the SRD), different art, in a different layout, at a different price point. And they can do so whenever they like, e.g., when WotC bails on 5e.
And then, when WotC move on to 6e and decide that that edition will be closed, Paizo will be stuck - the Pathfinder game will be gone and difficult to bring back, but they can't immediately use 6e, so they're in trouble.
Paizo won't be stuck because they'll still have PF2(5e).
ETA: you can bet Paizo is still discussing this.
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