The Iran war began with a joint US-Israeli attack -- though the interests of those two countries' leaders have since diverged -- while Lebanese militant group Hezbollah struck Israel shortly thereafter, sparking a devastating Israeli military response.

"Trump's strategy has been to compartmentalize the conflicts," said Sina Toossi of the Center for International Policy. "The problem is that Iran has never accepted that separation."

The US president is striving to negotiate with Iran, prevent a wider regional war, stabilize energy markets and contain the crisis in the Strait of Hormuz -- all while allowing Israel to pursue its military campaign in Lebanon, Toossi said.

Tehran has demanded that Lebanon be part of any agreement to end the war with Israel and the United States, which began on February 28.

Iran and Israel resumed striking each other over the weekend -- albeit on a limited scale -- for the first time since a fragile ceasefire took effect on April 8, with Tehran saying it was retaliating because Israel struck Beirut's southern suburbs, a Hezbollah stronghold.

Trump had urged Israeli Prime Minister Benjamin Netanyahu not to retaliate, seeking to avoid jeopardizing peace efforts the US president says are nearing fruition, but Israel hit back anyway.

Lebanon was dragged into the Middle East war on March 2 when Hezbollah fired rockets at Israel in retaliation for the killing of Iran's supreme leader Ali Khamenei.

Israel has since carried out a massive campaign of strikes that have killed more than 3,600 people in Lebanon, while also occupying a large chunk of the country's south.

Elisa Ewers, of the Council on Foreign Relations, said that Trump's effort to delink Lebanon from Iran "has not been all that successful."

"Iran is testing President Trump's resolve by continuing to demand that Lebanon be part of any initial negotiation," and "testing whether Trump would continue his backing for Israeli strikes," Ewers said.

She also noted that "Iran is trying to preserve Hezbollah capabilities as much as possible."

- 'Irony' -

The United States has hosted four rounds of Israel-Lebanon talks in Washington -- the first direct negotiations in decades between the two countries, which do not have diplomatic relations.

From the beginning, Washington has insisted on keeping the two peace tracks -- Lebanese and Iranian -- separate.

The talks have so far failed to end the violence, despite ceasefire announcements that have been rejected by Hezbollah and ignored by Israel.

According to Toossi, Tehran wants to demonstrate that "regional stability cannot be separated from Iran's security concerns or those of its allies."

The relationship between Trump and Netanyahu has meanwhile become increasingly tense.

"The real tension is that Israeli actions in Lebanon repeatedly risk undermining Trump's larger objective of stabilizing the region and reaching a broader 'end-of-war' framework deal with Iran that can reopen transit through the Strait of Hormuz," said Toossi.

Richard Haas, the former president of the Council on Foreign Relations, sees a certain "irony" in the situation, given that Trump launched the war against Iran at Netanyahu's urging.

"The 'peace' is near certain to drive a wedge between the two leaders and the two countries because it is unlikely to deal meaningfully with many of the issues Israel cares most about," Haas wrote in his newsletter last week.

Mona Yacoubian, of the Center for Strategic and International Studies, said that "ultimately, the Lebanon and Iran conflicts will proceed along separate tracks."

"While interconnected, each conflict is driven by its own set of dynamics related to conditions of the ground," she said.

"Unfortunately, this intersection more often will result in unpredictable conflict escalation transmitted from one arena to the other, rather than mutually reinforcing de-escalation," Yacoubian added.