The Islamic State in Iraq and Syria might have given Iran and Saudi Arabia a common enemy but that alone is unlikely to put a stop to their cold war. Just after the foreign ministers of both countries met in New York, Iranian-backed Houthi rebels in Yemen stormed through the capital city Sanaa and threatened to unseat the government there that is allied to Saudi Arabia.

Mohammad Javad Zarif, Iran’s top diplomat, spoke hopeful words in New York, saying, “Both my Saudi counterpart and I believe that this meeting will be the first page of a new chapter in our two countries’ relations.”

As the Reuters news agency points out, there were also hints of a detente last month when the Middle East’s rivals both welcomed the departure of Iraqi prime minister Nouri al-Maliki and his replacement by the more conciliatory Haidar al-Abadi. While the Saudis considered Maliki an Iranian ally, even the mullahs in Tehran recognized that his exclusion of Iraq’s Sunnis from power had given rise to the Islamic State’s insurgency which threatened their close ties with Baghdad. According to Reuters, “Once Iran came to see Maliki as too divisive and withdrew its backing, it removed a thorn in relations with Riyadh.”

But the agency recognizes much mutual suspicion remains. Iran sees the ruling family in Riyadh as stooges for their American foes and hasn’t forgotten their support for Iraq in its eight-year war with Iran in the 1980s. Saudi Arabia, for its part, still fears that Iran’s leaders remain determined to export their 1979 Islamic Revolution, “not least to Lebanon or the wealthy Sunni-ruled monarchies of the Gulf.”

The foreign policies of both countries have become more sectarian. Saudi Arabia competes for leadership of the Sunni world with Qatar and Turkey and supports Sunni insurgents in Iraq and Syria. Iran backs the Shia opposition in Bahrain, a close Saudi ally, and the Houthis in Yemen.

The Houthi belong to a particular offshoot of Shia Islamic and have important religious differences with the Twelver Shiism the Iranian regime adheres to. But as The American Interest notes, “sectarian distinctions within Shiism have certainly not stopped Iran from backing Syria’s Alawite-dominated Assad regime.”

The Iranian-Saudi rivalry, then, is not purely sectarian. It is as much about interests. The Iranians worry about being almost surrounded by an informal alliance that ties together Azerbaijan, Egypt, Israel, Jordan, Turkey, Saudi Arabia and the other Gulf monarchies — perhaps even Afghanistan. The Saudis are apprehensive about Iran disturbing the status quo where-ever it can: in Iraq, by propping up a Shia government in Baghdad, in neighboring Gulf states, by fueling Shia discontent, and in Yemen, by supporting the Houthi uprising.

The New York Times reported back in 2012 that “a relatively small but steady stream of automatic rifles, grenade launchers, bomb-making material and several million dollars in cash” was flowing from Iran into Yemen. That steady support now appears to have given the Houthis the upper hand. Al Jazeera reports they have seized much of the capital, “capping a decade-long uprising against the government.” The prime minister reportedly resigned. Some military units appear to have sided with the rebels.

The surprise Houthi offensive could set off a struggle with the dominant Sunni Islamist party in Yemen, al-Islah, which, according to Al Jazeera, they have identified as their arch-enemy — bringing the sectarian war that engulfs most of the Middle East to Yemen as well.

Iranian-Saudi rapprochement, it seems, will have to wait.