SEOUL — There are a handful of never-ending stories on the Vatican beat, dramas that flare up and recede without reaching resolution. Reconciliation between the Vatican and the traditionalist Lefebvrist movement that broke away 25 years ago falls into the category, as well as ecumenical dialogue with the Russian Orthodox.
Pride of place goes to relations with China, with the latest edition of the one step forward, one step back dance between Rome and Beijing playing out while Pope Francis is close by this week, in South Korea.
At the moment, there's little reason to believe the mini-ferment about the pope flying over China is an exception to the rule about possible breakthroughs, which amounts to "don't hold your breath."
The Vatican currently has diplomatic relations with 180 of the 193 states recognized by the United Nations, and China is the biggest prize among the remaining holdouts. Others include Vietnam, North Korea, Oman, and Saudi Arabia.
Rome has made its eagerness clear. One complication is that the Vatican has a diplomatic mission in Taiwan, the only European country that still does, but a former Vatican Secretary of State once said the Vatican would move to Beijing "overnight" if the green light would come.
Like clockwork, every six months or so there's a reported thaw, usually followed by a retrenchment that takes things back to the point of departure.
In 2007, for instance, Pope Benedict XVI issued a letter to Chinese Catholics, which was hailed as a potential turning point for its reassurances that the church had no ambition to "change the structure or administration of the state." He also revoked papal edicts that had barred faithful from contact with the state-sponsored Catholic body.
Not long afterward a bishop was appointed jointly by the Vatican and the state-sponsored Catholic group, which many saw as a harbinger of détente.
By 2010, it was back to business as usual, with China ordaining a bishop without papal approval, even sequestering three bishops loyal to Rome for days before the ceremony to force them to participate.
This week's version of the back-and-forth kicked off Thursday when Francis became the first pontiff to fly over Chinese airspace , dispatching a one-line telegram to President Xi Jinping that invoked "divine blessings of peace and well-being" on the nation.
Some commentators even fastened on the fact that Francis did not use the word "God," seeing it as a further opening to China's officially atheistic regime. (A Vatican spokesman cautioned against investing too much word-by-word "exegesis" in the message.)
In a reply faxed to news agencies, China's foreign ministry acknowledged the pope's telegram and said its government is willing to work with the Vatican to improve bilateral relations.
The fact that both the Vatican and China are under relatively new management, and that Francis and Xi have exchanged correspondence, added to the hopes for movement.
The wait wasn't long, however, for the other shoe to drop.
News broke just after Francis's arrival in South Korean that a small number of Chinese faithful who had been hoping to attend a gathering of Asian youth with the pope were unable to make the trip. Although reports conflicted about whether they had been formally detained or simply discouraged, a spokesman for the Catholic church in Seoul expressed "regret" and "concern."
"Maybe it's because of the Chinese local situation or some complicated situation in China," Father Heo Young-yeop said, adding he wouldn't comment further out of "fear for the safety" of Chinese youths already in Korea when they return.
If that's what improving bilateral relations looks like, the smart money isn't on rapid progress.
Given the headaches, one might wonder why the Vatican bothers. There are four core reasons it will likely press on.
• China is an emerging superpower, and the Vatican has no more desire to miss the train than anyone else. Rome is tenaciously committed to its unique status as a sovereign state and a player in global diplomacy, and opening relations with China would put an exclamation point on its role.
• The Vatican is concerned about the well-being of nearly 13 million Catholics in China, many of whom periodically face harassment and intimidation. Anything that might bring those folks solace would be seen with favor in Rome.
• The Vatican has always been alarmed about the split between an official church tolerated by the government and an underground community, because it creates the prospect of schism. The breakaway faction includes illicitly, yet validly, ordained Catholic bishops — meaning clergy Rome is constrained by its own theology to recognize as real bishops, even if they lack the pope's permission to act as such.
Almost nothing stirs Vatican nightmares like schism, and any pope would feel compelled to bend over backwards to try to end it.
• China is the last great missionary frontier on earth, with a burgeoning population, a deep spiritual hunger, and no dominant religious tradition. Vatican calculations are that the 13 million Catholics in China could easily become 130 million within a generation if there were an opening.
Some religious orders and Catholic movements periodically hold closed-door conferences in Rome to strategize for the "evangelization of China," just waiting for a political deal that would clear the ground.
As this week illustrates, there may be no firm basis to believe an opening is just around the corner. Fortunately, however, both Rome and Beijing are accustomed to taking the long view.
John L. Allen Jr. is a Globe associate editor, covering global Catholicism. He may be reached at email@example.com. Follow him on Twitter @JohnLAllenJr and on Facebook https://www.facebook.com/JohnLAllenJr. Sign up to be notified when Crux, a website covering all things Catholic, launches.