The short answer is no. Mainly because the Iranian hostages were released in 1981. The thread you link to has it all wrong. The hostage issue relates to the 1980 US presidential election, when Reagan faced the incumbent Jimmy Carter.

So, did Reagan make a deal with Tehran to ensure that the hostages weren't released during the election campaign? Again, the answer is a firm 'no'. There is, however, considerable evidence about what did happen.

Firstly, we need to consider the domestic Iranian situation. This is crucial, because in many popular accounts, the hostage crisis is framed solely in terms of US interests and US electoral politics. This gets it all wrong and ignores a crucial aspect of the entire story.

The hostages were taken because of the Shah's arrival in the United States (he was dying of cancer at the time) and became a symbol of anti-US resistance and an important tool within domestic politics.[1] If we look at what was going on in Iran months after the hostages were taken, by the summer 1980, the Ayatollah Khomeini had achieved many of his domestic political goals, for which the hostages had been an important lever (the radicalism of the students who had initiated the US embassy takeover had been co-opted by Khomeini against his more liberal opponents). Liberals within Iran's revolutionary political system had largely been dispensed with, those who had failed to resolve the hostage crisis to Iran's advantage had been sidelined, and the Majlis (the parliament) was now dominated by hardline Khomeini supporters. Moreover, the Shah had by then died of cancer, removing one particularly contentious issue.[2] Recall that one of the things the hostages were held for was a proposed exchange for the Shah.

The Iranian leadership was therefore willing to engage in good-faith negotiations with the Carter administration on the subject of the hostages release. Tehran's terms were transmitted to the White House via a West German diplomatic intermediary. These were:

1. A return of the monies the Shah had hidden away overseas (a lot of money. Like, loads).
2. The release of billions of dollars of Iranian funds held in US banks (these had been frozen by governmental order as part of the US sanctions against Iran).
3. That there would be no future legal claims against Iran by the hostages.
4. That the US would agree to a binding commitment never to interfere in Iran's domestic affairs again. This goes back to the 1953 coup that installed the Shah in the first place.

There was general agreement in Washington that this was a good deal. Indeed, US Deputy Secretary of State Warren Christopher met in secret with Iranian representative Sadeq Tabatabai to hammer things out. Once this had been sorted, there was an expectation that an agreement would be announced in fairly short order and that the hostages would be released.[3] But this didn't happen. Why?

Because in September 1980, Iraq invaded Iran, kicking off the brutal, eight-year Iran-Iraq War. Many Iranians - including important figures in the Majlis - believed that the US had given Saddam Hussein permission to invade Iran and were now in no particular mood to cooperate with Jimmy Carter. Given that the Majlis had to agree to any deal, this was a bit of a stumbling block. They did eventually start debating the deal at the end of October, once the initial shock of the war had worn off, but changes to the nature and structure of it were demanded. It's important to note that historians have examined this issue of the US giving a 'green light' to Saddam and found it based in myth rather than reality.[4]

The Carter administration had hoped that everything would be resolved by the election day, but it was not to be. But things didn't end there. After being defeated but before Reagan took office, Carter redoubled his efforts to get the hostages home. Indeed, he was tireless in this. Algerian representatives acted as intermediaries in all of this, shuttling notes, queries, memos, and reports back and forth between Washington and Tehran. Even though planes were ready to fly the hostages home on inauguration day, it never quite happened because of the complexities of international banking transfers of billions of dollars and various other bits of minutae. The hostages did come home, and Carter deserves a lot of credit for that, but they landed back in the US after he had left office, allowing Reagan to bask in the glory.

Of course, we know that Reagan's administration did later engage in shady dealings with Iran. However, the notion that Reagan's campaign colluded with Tehran for political advantage in 1980 runs counter to the actual historical record.

Notes

[1] Odd Arne Westad, The Global Cold War: Third World interventions and the making of our times (Cambridge: Cambridge University Press, 2005), 296

[2] John Ghazvinian, America and Iran: A history, 1720 to the present (London: Oneworld, 2020), 339-340

[3] Ibid

[4] Hal Brands, ‘Saddam Hussein, the United States, and the invasion of Iran: Was there a green light?’, Cold War History, 12:2 (2012), 319-343

Indicative Further Reading

John Ghazvinian, America and Iran: A history, 1720 to the present (London: Oneworld, 2020)

Rashid Khalidi, Sowing Crisis: The Cold War and American Dominance in the Middle East (Boston: Beacon Press, 2009)

Lawrence Freedman, A Choice of Enemies: America Confronts the Middle East (New York: PublicAffairs, 2008)

Daniel Streiff, Jimmy Carter and the Middle East: The Politics of Presidential Diplomacy (Basingstoke: Palgrave Macmillan, 2015)