The solution to the banking crisis sweeping the globe would be for customers to stop withdrawing deposits. But the banking system and regulators would have to calm fears before that happens system-wide.

That's why there are growing calls for US authorities to guarantee all customer deposits, regardless of whether they're insured or uninsured. The US Federal Deposit Insurance Corporation insures deposits at eligible banks up to $250,000 per account. European countries operate similar programs. 

If regulators insured all deposits of all sizes, similar to what happened to Silicon Valley Bank and Signature customers after those banks failed, that could give customers confidence that their money is safe with regional banks.

Moody's on Friday said there is a "high likelihood" that federal regulators could invoke a systemic risk exception to protect all uninsured depositors at First Republic. But if regulators make an exception for just one more bank, that would mean rescuing every teetering bank.

US authorities may be reluctant to do that if there's a chance the system can work out its problems on its own. With a looming debt ceiling crisis and intense scrutiny about using taxpayer money to fund anything close to resembling a bailout, the Biden administration would almost certainly prefer an organic solution to the crisis.

There's some evidence that may be starting to happen.

Western Alliance and Charles Schwab, which both reported a large amount of withdrawals last week, sought to reassure customers and investors by saying deposits were relatively steady over the past few days and they remained sufficiently liquid -- which means they have enough cash to continue to fund their operations.

And a US official told CNN that deposits at small and midsize American banks have stabilized in recent days, with outflows either slowing, stopping or in some cases reversing. 

Credit Suisse's problems -- years in the making -- are unrelated to the recent deposit runs at US banks. But after UBS came to the rescue of Credit Suisse, the flood of deposit withdrawals from US regional banks eased and central banks tried to make dollars more available to keep banks lending, there's hope the current banking crisis fades and the world evades an economic downturn.

What's certain: This isn't over yet.

-- CNN's Matt Egan and Phil Mattingly contributed to this report