NATO is in a race against time to save itself from a major embarrassment

Most importantly, that nation claims that Sweden allows members of recognized Kurdish terror groups to operate in Sweden, most notably the militant Kurdistan Workers' Party (PKK). Sweden changed its terrorism laws earlier this year, making it a crime to be part of these groups, though it's still unclear if this is enough for Ankara.

Turkey also claims that the Swedish government has been complicit in far-right protests where people burned copies of the Quran outside the Turkish embassy in Stockholm. Most recently, Turkey has said it wants Sweden to act after Swedish lawmakers projected the flag of the PKK onto the parliament building in Stockholm in protest at Turkish President Recep Tayyip Erdogan's re-election on Sunday. A spokesperson for the Swedish parliament acknowledged that people had projected images onto the side of the building, but had no specific evidence about what was projected or who was responsible, according to Reuters.

Finally, there are concerns at how willing Erdogan is to describe himself as a friend of Putin's. Shortly before he was re-elected, he told CNN that he and Putin share a "special relationship."

NATO officials and people within the Swedish government are now becoming concerned that missing the July 11 deadline -- the date of its next official summit in the Lithuanian capital Vilnius -- would send a dangerous message to the alliance's adversaries. These include Russia, and although nowhere near the North Atlantic, North Korea and China.

"If it's missed, it tells people like Putin that there is a weak link in the Western alliance. It gives them time and space to cause trouble," one NATO diplomat told CNN. "That could be anything from cyber attacks to funding and encouraging more Quran burnings to cause division in Sweden."

An Eastern European diplomat told CNN that as well as "emboldening the enemies" of NATO, any delay risks "giving the the sense of Erdogan's power over the alliance." The diplomat added that "Erdogan will use the moment to squeeze every drop from this situation and will throw the ball to Sweden -- making them hostage of their (own) anti-terrorist laws."

Officials from most NATO states are optimistic that a deal can be done before July, but are aware it could come with a price attached. 

Multiple officials point to the way that Erdogan struck a deal with the European Union that saw it hand Turkey 6 billion euros ($6.4 billion) among other perks in exchange for Turkey hosting Syrian refugees who were en route to Europe. Erdogan, European officials have repeatedly said, knew that he had Brussels over a barrel as he could effectively "flood" Europe with refugees at will.

What could NATO allies give Erdogan that might get him to change his mind over Sweden?

For starters, Turkey wants the US Congress to approve its purchase of US-made F-16 fighter jets. While US officials are reluctant to tie the Sweden issue and F-16s overtly, officials say that behind the scenes there is an obvious deal to be done. 

Diplomats are also well aware that Turkey's economy is in dire straits, with soaring inflation and a collapse in the value of its currency against the dollar, and that both the US and EU currently have sanctions imposed on the country. 

While there is room for a deal to be done -- and the allies in favor of Sweden joining do have leverage -- there are a couple of issues that could see July 11 come and go without NATO getting it wants. 

The first is Erdogan's unpredictability. Sunday's election was the closest he has come to losing power in 20 years, which allies fear means he might double down on Sweden when it comes to its anti-terror policy.

Sweden is unlikely to introduce anything that looks as authoritarian as Erdogan would probably like to see in place, especially when it comes to the Kurds; at this point the only resolution could be that Erdogan claims the changes Sweden has already made to its terror laws as a personal victory and moves on. 

The second is that Turkey isn't the only fly in the ointment: Hungary also objects to Sweden joining NATO.

These two issues at some level interact with one another: if Erdogan were to accept Sweden's anti-terror laws as sufficient -- only for Hungary to block the whole thing risks making him look weak by comparison, European officials fear. 

For their part, the pro-Sweden allies -- including the United States and United Kingdom, arguably the two most influential NATO members -- are doubling down on July 11 and privately offering Sweden assurances that it is their priority, no matter what Turkey does. 

Sweden joining NATO would be the latest in a long list of good news stories for the alliance since Russia invaded Ukraine. Officials have been surprised at the level of unity in the alliance since the war began and have been delighted at renewed pledges on defense spending and strengthening the alliance. 

Russia launched its war in the first place partly due to NATO's expansion, a move that shows no sign of slowing, with Ukraine now also wanting to join the alliance. Even the Japanese are shifting towards NATO, with the country's foreign minister telling CNN earlier this month that it is in talks to open the first NATO liaison office in Asia. 

For all the talk of NATO facing what French President Emmanuel Macron called "brain death" not so long ago, it's undeniable that the alliance has a renewed sense of purpose and is confident about its future. That's precisely why officials are so concerned about Turkey vetoing Sweden's accession on NATO's own timetable. 

Just as a chain is only as strong as its weakest link, an alliance is only as united as its latest act of unity. In the modern world of diplomacy, signals and subtext matter a huge amount. And while it might seem insignificant exactly when Sweden does or does not join NATO, Turkey giving the alliance's enemies the faintest whiff that members can be picked off would, officials believe, upend months of good work that have brought the alliance closer together than at any other time in recent memory.