I think the best way to answer your question is to say: the premise is highly implausible, so the issue simply does not arise.
I might as well ask: if we suppose that we discover time travel, is there cause for alarm about password security? Sure, if we discover time travel, someone could travel back in time, appear poof just before I enter my password, look over my shoulder when I typed my password, and poof disappear before I notice.
Or, if my computer can time-travel on demand, I can set it up to iterate the following loop: pick a random password guess, try to see if it is correct, if it is correct print the password and halt; if it is incorrect travel back in time to the beginning of the loop and start over. If time-travel algorithm is possible, this is an O(1)-time algorithm to crack any password, given its password hash!
But of course these answers are silly, because for the foreseeable future there is no realistic likelihood of anyone discovering how to travel back in time. Similarly, for the foreseeable future there is no realistic prospect of someone discovering an algorithm to solve all SAT instances efficiently. Sure, if someone could find a SAT-solving algorithm that could solve every SAT instance in 15 seconds, then they could crack every password (given its password hash) in 15 seconds. But I don't think that's terribly likely to happen in my lifetime.
P.S. I see from clicking on your link that you would probably prefer a more technical answer. My suggestion is to read up on Hellman's time-space tradeoff and on rainbow tables; that will give you a better understanding of the state-of-the-art methods applicable to password cracking. You might also want to read up on why rainbow tables are not applicable to salted passwords; similar reasons are likely to apply to your methods.
Looking at the complexity of the method in your link, I see your method requires a 2v precomputation, where v is the number of variables in the SAT instance. In contrast, Hellman's time-space tradeoff and rainbow tables require a 2n precomputation, where n is the number of bits of the password (the number of bits input to the hash function). In the password setting, n is going to be much smaller than v, so rainbow tables and Hellman's time-space tradeoff look like they will perform better than your method. In other words, it doesn't look likely to me that your method -- even if it is valid -- will beat the state-of-the-art at password cracking or will have much relevance to password cracking in practice. Of course, you could always try it in a little experiment, and compare your method to existing password crackers; that would be the true test. But at the moment, I see no reason to expect advances in SAT solving that would be relevant to password cracking.